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MATHEMATICS | PORTFOLIOS
Understanding Volume and Area in Mathematics
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Math and our World
Written By :
Mr. Calvin Musk, CEO at Calvin Industries Cooperation
Date of Publish :
December 10th, 2022, 10:18 AM MST
Department :
CalvinX Cooperation at Calvin Industries Corp.
"Good mathematics is not about how many answers you know.. its about how you behave when you don't know".
We are in a constantly evolving state in mathematics, and we are only getting to understand more about numbers. Math is constantly evolving as we are getting exposed to new information. While many see mathematics as just numbers, word problems, and equations- to me, mathematics is the ability for a human to learn more about their environment. For example, mathematical models and algorithms can be used to analyze data from sensors and other sources to gain insights about the environment. Additionally, mathematical concepts and principles can be used to understand and explain natural phenomena and the behavior of objects in the world. For example, calculus can be used to model and analyze the motion of objects, while geometry and trigonometry can be used to understand the shape and size of objects and the relationships between them. Overall, math provides a powerful set of tools and techniques for understanding and analyzing data and information about the world around us, and it can be an invaluable tool for learning more about our environment. I don't want to see math as just numbers- I want to be able to see math for its capabilities in better learning more about us, our environment, and our surronding world. The Fibanacci sequence, The Fibonacci sequence has many interesting properties and appears in many different contexts in mathematics and other fields. One of the most famous applications of the Fibonacci sequence is in the study of the growth of populations of animals, such as rabbits, in which each pair of rabbits produces a new pair of offspring every month. In this context, the Fibonacci sequence can be used to model the growth of the population over time, and the ratio of successive numbers in the sequence approaches the famous golden ratio, which is approximately 1.6180339887. This ratio has been found to occur in many natural phenomena, including the arrangement of leaves on a stem and the shape of spiral galaxies. Overall, the Fibonacci sequence is a fascinating mathematical concept that has many interesting properties and applications, and it has been studied and studied for centuries. It continues to be an active area of research and continues to fascinate mathematicians and other researchers. While our original unit of patterns and relations have come to a halt to our transition to volume, shape, and area- I am interested in how math is applied to our real-world in better allowing us to explore our environment. Theres a connection between mathematics, and science, and not many is capable of seeing it.
At Calvin Industries, we dedicate ourselves to seeing mathematics beyond numbers, patterns, equations, expressions, and problems. We want to see math for its correlation to our world. In this unit, we will be discussing more about : area, volume, shapes of prisms and 2 dimensional shapes, and calculating mass, density, and volume based on our science unit.
Unit Overview
Written By :
Mr. Calvin Musk, CEO at Calvin Industries Cooperation
Date of Publish :
December 10th, 2022, 10:18 AM MST
Department :
CalvinX Cooperation at Calvin Industries Corp.
Welcome to shape, volume, and area. Through this unit, we will demonstrate how to calculate the area and volume of different 2D and 3D shapes and prisms, how they compare, and explore circles, circumference, and other related topics. We will be experimenting with a variety of shapes, such as squares, triangles, and circles- and the 3 dimensional version of that- cubes, cuboids, triangular prisms and pyramids, and cylinders and spheres. Before we begin, let us examine the difference between a 2D and a 3D shape. What is the difference between a two-dimensional and a three-dimensional one? Are there different units that we use to measure them?
A 2-dimensional shape is a flat shape and only has one face. We measure 2D shapes through area and perimeter. The perimeter is the total length of the boundaries within a shape, while the area is the amount of space within that boundary of our perimeter. Examples of 2D shapes are squares, triangles, octagons, pentagons, etc. They are classified as 2D shapes as they only have one face and are flat shapes that cannot support itself unlike 3D shapes. They are flat shapes and have no thickness. The significant difference is that 2D shapes can only stand up on themselves on a flat surface. A 2D shape is a geometric figure that is defined by its length and width. There are many different types of 2D shapes, including squares, rectangles, circles, triangles, and hexagons. These shapes are commonly found in everyday objects such as stop signs, pizza slices, and clocks. In addition to their practical uses, 2D shapes are also the building blocks for 3D shapes, which have length, width, and height. Unlike 2D shapes, which are typically flat, 3D shapes have volume and are more three-dimensional in appearance. However, even though 2D shapes are flat, they can still have depth and can be used to create optical illusions and other interesting visual effects.
A 3-DIMENSIONAL SHAPE. A three-dimensional shape, also known as a 3D shape or solid figure, is an object or figure that has three dimensions: length, width, and height. These dimensions give the shape its volume, or the amount of space it occupies. In contrast, two-dimensional shapes are flat and have no thickness or depth. All 3D shapes have volume and can be rotated to view their different faces and edges. The volume of a 3D shape is a measure of the amount of space it occupies and is calculated using the formula length x width x height = volume. Unlike 2D shapes, 3D shapes have a defined volume based on mass. Some common examples of 3D shapes include cubes, cuboids, cylinders, pyramids, prisms, and spheres. These shapes are often used in geometry and other mathematical fields to help visualize and understand complex concepts. They can be manipulated and studied from different angles, which allows us to better understand their properties and characteristics. Like discussed in Ms. Ivy's class, we can think of a 3D shape as the same as a 2D shape, but composed of many different layerings to make it 3 dimensional. It's like taking a square of 1 layer, and multiplying it to give it more than 1 face. Unlike 2D Shapes, 3D shapes are directed towards surface area rather than perimeter. We define 3D Shapes with volume, as they are composed of multiple areas and faces, while we define 2D Shapes with area because they only have 1 face.
Through the visual provided above, we can see that a 2D square only has 1 side, 4 verticies, and 4 edges. If we multipled that to compose a shape of many different dimensions, we have a 3 dimensional shape of a cube. A cube has 6 faces, 8 vertices and 12 edges. But, what are these terms?
The following images label and describe the different parts we can see in a 2D and 3D shape. We come across a variety of objects of various shapes and sizes in our daily activities. There are golf balls, doormats, ice cream cones, and coke cans, stop signs, spaceships, and many among other things. These objects have distinct characteristics such as length, breadth, diameter, and so on that distinguish them from one another. Regardless of how different their dimensions are, they all occupy space and have three dimensions. As a result, they are known as three-dimensional Shapes or solids. There are figures with two dimensions, length and breadth, that can be represented on a plane (as a piece of paper). They are also known as two-dimensional or plane figures. In this article, we will look at the meaning of faces, edges, and vertex in Math for solid objects. But, before we get into it, lets have a recap onto the terms related to dimensional shapes :
Edges : An edge in geometry is a line segment that connects two faces of a three-dimensional shape. Depending on its shape and complexity, a three-dimensional shape can have any number of edges. A cube, for example, has 12 edges, whereas a sphere has none. The concept of edges is irrelevant in two-dimensional shapes because there is no "depth" or "thickness" in two-dimensional shapes. Instead, sides are used to describe the various parts of a two-dimensional shape. A triangle, for example, has three sides, whereas a rectangle has four sides. Edges are a crucial concept in geometry because they define the shape and size of a three-dimensional figure.
Vertex : A vertex, sometimes called vertices, is a point in geometry where two or more lines or edges intersect. A vertex is a fundamental geometric shape building block, and the total number of vertices in a shape is a defining characteristic of that shape. A vertex in two-dimensional shapes is simply the point at which two sides of the shape meet. A triangle, for example, has three vertices, whereas a square has four. A vertex is a point in three-dimensional shapes where three or more edges meet. A cube, for example, has eight vertices, whereas a pyramid has five. Vertices are important in geometry because they define the shape and size of a geometric figure.
Face : A face is a flat surface on a 2D or 3D shape. For example, a cube has a total of 6 faces. A cylinder has a total of 2 faces, because it has 2 flat surfaces- one at the top, and one at the bottom at overall connects it together. The term "face" in geometry refers to a flat surface of a three-dimensional shape. Depending on its shape and complexity, a three-dimensional shape can have any number of faces. A cube, for example, has six faces, whereas a cone only has one. Faces aren't as widely used to portray two-dimensional shapes because there is no "depth" or "thickness" in two-dimensional shapes. Instead, sides are used to describe the various parts of a two-dimensional shape. A triangle, for example, has three sides, whereas a rectangle has four sides. Faces are crucial in geometry because they are used to calculate the surface area of a three-dimensional shape. To determine the surface area of a shape, we must first determine its area.
Perimeter : The perimeter is the total length of the boundary of a shape, and the total distance that surrounds -the outside of a shape like the edges. In order to find the perimeter of a shape, we need to find the sum of all of the edges combined. The perimeter of a two-dimensional shape is the distance around its outside. It is usually expressed in linear units like feet, inches, or metres. To find the perimeter of a shape, add the lengths of all its sides together. To find the perimeter of a rectangle, for example, add the lengths of all four sides of the rectangle. If the rectangle's length is 5 feet and its width is 3 feet, the perimeter of the rectangle is 5 + 5 + 3 + 3 = 16 feet. Perimeter is not a relevant concept in three-dimensional shapes because three-dimensional shapes do not have a "outside" or "edge" in the same way that two-dimensional shapes do. Usually in 3D shapes, we refer perimeter with surface area instead.
Area : The area is the total amount of space within the boundaries of the perimeter. How much total space does the shape take up? How big is the football field? These are common questions associated with locating the area of a shape. Area is mainly observed in 2-dimensional shapes, and is used to measure the face within a 3 dimensional shape. Area is a measure of the size of a surface or shape. It is typically measured in square units such as square meters, square feet, or square inches. To find the area of a shape, we need to use the appropriate formula for that shape. For example, the area of a rectangle is calculated by multiplying the length and width of the rectangle, while the area of a circle is calculated by multiplying the radius of the circle by itself and then by the value of pi (approximately 3.14). Area is an important concept in geometry and is used to measure the size of two-dimensional shapes such as triangles, circles, and squares, among others. It is also used in everyday life to measure the size of objects or surfaces, such as the size of a piece of land or the size of a carpet. In general, to find the area of a shape, we need to know the dimensions of the shape and use the appropriate formula to calculate the area. For example, to find the area of a rectangle, we need to know the length and width of the rectangle, and then use the formula area = length x width to calculate the area. Similarly, to find the area of a circle, we need to know the radius of the circle and use the formula area = pi x (radius x radius) to calculate the area.
Volume : Volume is mainly seen in 3-dimensional shapes that measure the total amount of space that the shape occupies. When measuring volume, we usually measure volume through cubic meters. When we get a product that correlates to volume, we are literally finding how many pieces it takes to fill up our shape. Imagine having a container, and needing to fill it up with shapes. That is technically what volume is. But, It is also important to note that volume is an intensive property of a substance, which means that it is not affected by the amount of the substance that is present. For example, whether you have a cup of water or a swimming pool full of water, the volume of water will be the same, even though the amount of water is different. How we find the volume of a shape will depend directly on the the shape, which will be mentioned below.
Circumference : The circumference of a circle is the distance around the outside of the circle. Read below for more information on revolving circumference, and how we can use information revolving diameter and radius to identify the circumference.
Radius V.S Diameter : Many make mistakes revolving radius and diameter when calculating the circumference. Radius is the point from the edge of the circle, to the center point, where diameter is the point from the edge of the circle, to the opposite edge. When calculating circumference, it is vital to calculate using the radius rather than diameter. If we have the diameter and we want to get the radius, we can simply divide the diameter by 2. If we have a radius and want to get the diameter, we can simply multiply the radius by 2. The radius is a key measurement of a circle, and it is used in many mathematical formulas and calculations involving circles, therefore, it is critical to not get them mixed up.
Cross Section : Judging by its name, the cross-section is when we cut through a shape directly through the middle. Through shapes like the cylinders, rectangular prisms, and etc, we can create symmetrical shapes through this process. When we cut a shape in half and we get symmetrical sides, then this shape is known as a prism.
Pie : The number pie, also known as pi, is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. The value of pi is approximately equal to 3.14, although it is an irrational number that goes on infinitely without repeating. Pi is an important constant in many mathematical formulas and calculations involving circles, spheres, and other curved shapes. It is also widely used in engineering, physics, and other fields.
Surface Area : The surface area of an object is the total area of its outer surface. It is a measure of how much space the object occupies in two dimensions. For example, the surface area of a cube would be the total area of all six of its sides. The surface area of a sphere would be the total area of its curved surface. Surface area is often used in geometry and physics to calculate the amount of heat, light, or other forms of energy that an object can absorb or emit. It is also a useful measure for comparing the size or volume of different objects. In general, an object with a larger surface area will have more space available for interactions with its environment, such as chemical reactions or the exchange of heat or light.
Now that we have a basic understanding revolving the concepts of volume :
Q ) Explain how you could find a missing dimension for a prism when you know its volume.
Before we begin, what even is a prism? A prism is a shape that if we directly split it through the middle, the two pieces will be identical to each other. An example of a prism is a rectangle, cylinder, etc.
To find a missing dimension for a prism, you must first understand the formula for prism volume, which is the product of the prism's base area and height. Depending on the shape, there may be different formulas associated to it. For example, a rectangular prism is base times height times width, where a triangle is base times height times width divided by 2. Its important to understand the shape and formula associated with the shape before we move on. If we have an understanding of the volume, we must find different factors that may equal that amount. The base area is the area of the shape that forms the prism's base, and the height is the distance between the base and the prism's top. For instance, if you know the volume of a rectangular prism is 100 cubic units and the dimensions of its base, you can use the formula to solve for the missing dimension, which is the prism's height- through subsitution. The volume of a rectangular prism is calculated as V = l * w * h, where l is the length of the prism, w is the width, and h is the height of the rectangular prism. However, if we only know the volume, there may be a lot of different factors that may amount to that volume, but, if we are given more factors, we can better narrow it down. To find the missing dimension, rewrite the formula to account for the unknown variable (in this case, the height). This gives you the equation h = V / (l * w), where V is the prism's known volume, l is the prism's known length, and w is the prism's known width. Once you have this equation, you can plug in the known values for V, l, and w to find the missing dimension, which is the rectangular prism's height. For example, if the volume of the rectangular prism is 100 cubic units, the length of the base is ten units, and the width of the base is five units, you would enter these values into Ththe equation to find that the height of the rectangular prism is 2 units.
Basically, just substitute the variables with what you know. The length, width, height, and in this case, the volume. Using opposite operations, we may be able to narrow it down to a few possible dimensions and factors that equal to our volume. For example, if you know the volume of a rectangular prism and the length and width of its base but not its height, you can use the volume of a rectangular prism formula to create an equation containing the known dimensions and the unknown height. The volume of a rectangular prism is calculated as V = l * w * h, where l is the length, w is the width, and h is the height. This answer also amounts to explaining how you could find the volume for any prism, with any kind of cross section. We can do this by using the formula of the shape, like base times height times width, to find the base and volume.
Square, Cubes, and Cuboids
A square is a two-dimensional shape that has four equal-length sides and four corners, or vertices. It is a regular polygon, which means that its sides and angles are all equal. A cube, on the other hand, is a three-dimensional shape with six identically sized and shaped square faces. It is also a regular polyhedron, with the same regular shape on all of its faces.The primary distinction between a square and a cube is their dimension. A square has two dimensions, whereas a cube has three dimensions. A square exists on a flat surface, whereas a cube has length, width, and height. Furthermore, unlike a square, a rectangle has four sides and four corners.
Rectangles, on the other hand, is not considered a square because of how its not polygon shape. Though it has 4 edges and 4 vertexes, it is not considered a square because of that. The 3 dimensional form of a rectangle is known as a rectangular prism, and shares the same characteristics as a cube.
A square is a shape where all the sides are equal, and the corners are at 90 degrees ( right angle).A rectangle is a shape where 2 sides in opposition are equal, and the other 2 sides in opposition are equal- meaning we have 2 separate perimeters. Similar to a square, a rectangle also has 90 degree right angles per corner. Technically, every square is a rectangle, but not every rectangle is a square. Similar to square and rectangle, they have 4 sides, and 4 vertices.
Measuring 2D Squares
In order to measure a 2-dimensional square, we need to know its height and width. The same goes for a rectangle- they all follow the same formula of length times height. Luckily, for a square, all the sides must be the same for it to be considered a square. Gather our width and length, and for the example I provided, the width and length of our shape is all 5 centimeters. 5 times 5 is 25, which is the area of our square. To find the perimeter, we just have to add all of our sides together. We have 4 sides, each measuring 5 centimeters. Adding up our perimeter, we can start off with 4 for our corner. Then, we add the ones in between our corner pieces. 3 + 3 + 3 + 3 equals 12, and adding our corner pieces, we get a total perimeter of 16.
With a rectangle or a cuboid, its the same concept. But, all the sides likely won't be the same, so, its important to have an understanding of the values prior to doing the equation.
A cube is a shape where all the sides are equal- length, width, and hight, and where all the corners are a right angle, but, rectangles aren't the same. They do share very similar characteristics, however- like how they both have 6 faces, they both have 8 vertices and 12 edges, making them technically the same shape. But, because rectangles and their edges aren't all the same width and length, rectangles are a special type of cube.
Measuring 3D Squares
In order to measure a 2-dimensional square, we need to know its height and width. The same goes for a rectangle- they all follow the same formula of length times height. Luckily, for a square, all the sides must be the same for it to be considered a square. Gather our width and length, and for the example I provided, the width and length of our shape is all 5 centimeters. 5 times 5 is 25, which is the area of our square. To find the perimeter, we just have to add all of our sides together. We have 4 sides, each measuring 5 centimeters. Adding up our perimeter, we can start off with 4 for our corner. Then, we add the ones in between our corner pieces. 3 + 3 + 3 + 3 equals 12, and adding our corner pieces, we get a total perimeter of 16.
With a rectangle or a cuboid, its the same concept. But, all the sides likely won't be the same, so, its important to have an understanding of the values prior to doing the equation.
Surface area measures the total amount of area that a 3 dimensional shape covers. We can measure the surface area of a cube by multiplying the area of one face by 6, since a cube has 6 faces, and we can measure the surface area of a rectangular prism by just adding up the total area of each of the 6 shapes to get the total surface area.
Parallelograms
Parallelograms are special types of rectangles that share common characteristics between themselves and rectangles. In order to find the area of Parallelograms, it's the same formula as you would see in a square and rectangle. Its name derives from how it is parallel on both sides. They share common characteristics between itself and rectangles on how it has 4 vertices, 4 edges, and how 2 sides are equal in opposition to each other, and how the other 2 sides are also equal in opposition. The major observable difference between the two is that a parallelogram has opposite sides equal, while in a rectangle, the opposite sides are equal with all its adjacent sides being perpendicular to each other.
The Painted Cubes Problem
The average 3 x 3 rubik's cube is composed of 27 different cubes. We can use linear relations to express the movement and transition of our cube -
3 Faces : This value will always remain at 8 pieces. Only 8 pieces, aka the corner pieces, will have 3 faces covered in paint. No matter how big our cube is, 69 x 69, 420 x 420, we will always only have 8 corner pieces.
2 Faces : On a regular 3 x 3 cube, 12 of the cubes will be covered in paint- and these are the cubes along the perimeter of the cube, excluding the corner pieces. As we increase the rubiks cube, the value will increase by 12. each time, because we are adding a new piece to the boundries of the cube.
1 Faces : Any cubes that are within the boundries and make up the area rather than the perimeter will only be covered on one of its face. As we increase the size of our Rubrick's cube, the pieces that are within our cube and within the boundries will have 1 face covered. Similar to the one prior, the cubes with only 1 face covered will always be a multiple of 6. The number increases by 6, 18, then 30 with a difference of 12.
0 Faces : In a 3 x 3 cube, only 1 cube has no faces covered in paint- and that is the cube in the centre, un-exposed to the outside world. However, as we increase the size of our Rubicks cube, more will be contained within our Rubick's cube, and will be exposed to no paint.
The Cuboid Problem
Originally, our shape was a 20 centimeter by 20 centimeter grid, but, since we cut the corners of our box, that has turned into a 18 x 18. If we fold the flaps to form a box without a lid, we would get a 18 x 18 x 1- 18 for our width, 18 for our length, and 1 for our height. This would mean the volume of our box is around 324 centimeters/3. To get the maximum amount of volume, we must use the equation that Google has provided to find volume : x(20-2x)^2 = v(x)?. But, because I have no idea what this means, we are not going to follow this exact equation. Rather, we can use what we already know. You can vary the size of the squares you cut off to make boxes with different volumes. For example, if you cut 5 cm by 5 cm squares, the volume of the box will be 20 cm x 20 cm x 5 cm, or 2000 cubic centimetres. If you cut off squares 7.5 cm by 7.5 cm, the volume of the box will be 20 cm x 20 cm x 7.5 cm, or 3000 cubic centimetres. In general, the larger the squares you cut off, the smaller the volume of the box, and vice versa. Using this idea, the maximum possible volume of this type of box that can be made is 512/cm. When we cut more of the corners, although the total area may decrease, the total height will increase, resulting in a higher product. Therefore, if we cut the corners by 2 centimeters, we will get 16 times 16 times 2 which is equal to 512 cm.
Question ) A rectangular prism has a total volume of 12 units3 . What could be the possible side lengths of the prism?
This is quite a simple question where all you need to do is find 3 factors that equals to 12. Some possible answers include 1 x 2 x 6, 3 x 2 x 2, 4 x 1 x 3, 6 x 1 x 2, 3 x 1 x 4, and etc.
Triangles and Triangular Prisms
A triangle is a polygon with three sides. It is a geometric shape with three straight sides that are joined at three vertices, or corners. The three sides of a triangle are commonly known as the base, height, and hypotenuse. Triangles can be classified according to their sides or their angles. An irregular triangle has sides that are different lengths, whereas an equilateral triangle has all three sides that are the same length. An isosceles triangle has two sides that are the same length. Triangles can also be classified according to their angles. If two of the three angles are the same, the triangle is called an equiangular triangle. But if 2 angles are the same, it is called an isosceles triangle. If a triangle has one angle that is larger than the other two, it is called an obtuse triangle, but if all three angles are less than 90 degrees, it is called an acute triangle. Triangle are usually identified through their 3 edges and 3 verticies that connect them together. Triangular prisms are basically like an extended triangle, where they have 5 faces, 12 edges, and 9 vertices.
Did you know this about triangles? Here are some interesting facts I found on the web about triangles : The delta symbol, which is frequently used in mathematics to represent the area of a triangle, is named after the Greek letter Delta, which is shaped like a triangle. The mathematical constant pi, which is the ratio of a circle's circumference to its diameter, can be calculated using triangle properties. One of the most famous theorems in mathematics is the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. The golden ratio, which is approximately 1:1.61, is frequently used in art and architecture to create aesthetically pleasing compositions.
Measuring 2D Triangles
The formula for calculating the area of a triangle is: area = (base x height) / 2. This formula applies to any triangle, regardless of its side lengths or angle sizes. This formula works because it exploits the fact that the area of a triangle is equal to half of the product of the base and the height. This is due to the fact that a triangle is a two-dimensional shape, and the area of any two-dimensional shape equals the product of its base and height divided by two. This is true for triangles because their bases and heights are perpendicular to each other, just like the base and height of a square.This equation works because we can see a triangle as half of a square/rectangle that's split in half. We can use my example of a triangle with a length and height of 5 centimeters. 5 times 5 equals to 25, and divide that by 2 to get a total of 12.5 cubic centimeters which is the total area of that triangle. For example, if the base of the triangle is 10 units and the height is 5 units, the area of the triangle would be: Area = (10 x 5)/2 = 25 units^2
Did you know this about triangular prisms? A polyhedron with two triangular bases and three rectangular faces is known as a triangular prism. A triangular prism's faces are parallelograms, which means that opposing faces are parallel and congruent. The lateral faces are the three rectangular faces of a triangular prism. The triangular bases of a triangular prism are referred to as the prism's base and top. A triangular prism's edges are perpendicular to the bases and connect the corresponding vertices of the two triangles. The prism's median lines are the line segments that connect the midpoints of the sides of the triangular bases. The altitude of a triangular prism is the distance between its base and its top.
Measuring 3D Triangles
Measuring triangles is very similar to which we would measure the volume of a cube or cuboid. Again, this is because triangles/triangular prisms are technically half of a cube. Imagine grabbing a cube, and cutting it in half diagonally- we have the product of a triangle. We can use the formula b x h x w / 2 to find the area of a triangular prism. Take my example. We have the base of 5 centimeters, the height of 5 centimeters, and a width of 5 centimeters. 5 x 5 x 5 gives us a product of 125, and divide that by half. 125 / 2 gives us a product of 62.5 cubic centimeters, which is the total area of our triangle. According to BEDMAS, multiplication and division is part of the same order, so, we don't need to worry about which operation to do first. We can do 5 times 5 / 2 times 5 and still get the same product, but, this may be different varying on the different triangles. We can envision a triangular prism as a regular 2D triangle with multiple layers. Its like finding the area of the face within a triangle, like 5 times 5 / 2 to get 12.5 to resemble the face of our triangle, and multiplying it by the amount of layers (5). 12.5 times 5 gets us 62.5 cubic centimeters.
Pyramids
Pyramids, unlike triangular prisms, are types are triangles where the edges intersect at the top and all meet at a common point. A pyramid is a polyhedron that has a polygonal base and triangular faces that meet at a common point called the "apex". The base of the pyramid can be any polygon, but the most common pyramid is a square pyramid, which has a square base. The simple formula for finding the volume of a pyramid is to time base by height and width, and dividing the value by 3 to get the total volume
Question ) A triangular prism has a volume of 300 cm3.
Which could not describe its base and height?
A Triangle base = 15 cm and triangle height = 4 cm, height of prism = 10 cm
B Triangle base = 6 cm and triangle height = 5 cm, height of prism = 20 cm
C Triangle base = 5 cm and triangle height = 2 cm, height of prism = 30 cm
D Triangle base = 4 cm and triangle height = 3 cm, height of prism = 50 cm
Answer ) Option C cannot describe the base and height of a 300 cm3 triangular prism. In this case, the triangular prism's base would be a triangle with sides of 5 cm and 2 cm, resulting in a base area of (5 x 2)/2 = 5 cm2. The triangular prism would be 30 cm tall, with a total volume of (5 x 30) = 150 cm3. Option C cannot describe the base and height of a triangular prism with a volume of 300 cm3 because this is less than the given volume of 300 cm3. However, the other options - a, b, and d- is possible of giving an equal volume to 300 cm. We know that in order to find the volume of a triangle, we must multiply the b x w x h and divide it by 2. As the product in c is less than that given amount, this is not possible considering we would need to half it. However, option d gives us a product of 600, which where we divide it by 2 will give us the volume we are looking for of 300 cubic centimeters.
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Question ) What measurements of the triangle base do you not need to know to calculate the volume of a triangular prism?
Answer ) In trying to find the volume of a triangle, we usually times base times height times width, meaning that the side edges aren't needed in calculating the volume of a triangular prism. However, knowing the lengths of the sides can be useful in calculating other properties of the triangle, such as the area or the perimeter. The side lengths of the hypotenuse, etc.
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Question ) What might be the dimensions of a triangular prism if the prism’s volume is 1 m3 and its height is 12.5 cm?
Answer ) First, lets identity how many centimeters is equivalent to one meter. That would be 100. 100 centimeters would be equivalent to 1 meter. For better referencing, we can reword the word problem so we can easily compare and contrast our values. When identifying the volume of a triangular prism, we need to find 3 different factors that equal to our total volume- doubled. If we were finding the volume of a rectangular prism, we would simply just find 3 factors that equal to that volume, but, with triangular prisms, its the same thing, but, the final product needs to be divided by 2 since a triangle is a diagonally cut rectangle. If we double 100 centimeters to 200 centimeters, we would just need to find 3 factors that give us the product of 200. We already have one of the dimensions stated, so, we can create the equation 12.5 x (substitute) x (substitute) = 200 cubic centimeters. There are multiple different possibilities onto what the answer may be, but, its important to remember that the product of these 3 factors won't get you 100 centimeters alone- you must divide it by 2 in accordance to area = b x h x w / 2.
12.5 times 2 times 8 : This brings us a total of 200 centimeters and could be a possible dimension. Divide 200 centimeters by 2 to get 100 cubic cm.
12.5 times 4 times 4, 12.5 times 5 times 3.2, 12.5 times 6 times 2.666666666666666666, 12.5 times 8 times 2, 12.5 times 10 times 1.6, etc.
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Question ) Could the volume of a triangular prism be 100 cm 3 ? Explain.
Answer ) Well, the volume of a triangular prism can be pretty much anything! 100, 69, 420, 1,000. No matter how big, or small the aimed value may be- there will always be a factor that equals to that amount. The volume of a triangular prism is calculated by multiplying the triangle's base area by its height. For example, if the area of the triangular prism's base is 25 square centimeters and its height is 4 centimeters, the volume of the prism is 100 cubic centimeters, which is the product of 25 and 4. A triangular prism can also have a volume of 100 cubic centimetres if the base of the triangle has an area of 10 square centimeters and the height is 10 centimeters, or if the base has an area of 50 square centimeters and the height is 2 centimeters, and so on. Even a triangular prism can have the volume of 0.69 cubic centimeters, where each dimension could equal to 0.46.
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Understanding Triangular Prisms
In order to calculate the volume of this triangle, we must following the formula base times height times width divide by 2. In this case, the length is 30 centimeters, and the rest is a bit hard to tell. But, because the order in which we do multiplication does not matter, the order in which we multiply the dimensions aren't really important in finding the volume. It doesn't matter if we multiply height, width, or length first- because at the end of the day, we will get the same answer, unlike division and subtraction where the order matters. We can do 30 times 20 times 10, which gives us a total of 6,000. Divide that by 2 to get the total area of 3,000 cubic centimeters as our total area. If the total height were to increase to 40 centimeters, then that would overall need to change the entire equation from 10 x 30 x 20 / 2, to 10 x 40 x 20, which would then give us a greater product of course. Our original product was 3,000, but if we changed the height to 40, that would change our product over to 4,000 cubic centimeters. The height in this case is demonstrating how thick or stretched out our triangular prism is, and if we increase that, we would have to multiply the total area by a value greater. The faces that lies above and below the shape each represent about 200 squared. We would need to multiply the total area of the 2 dimensional shape by the amount of layers there are, to increasingly double the total volume. 200 times 30 / 2 gave us an original value of around 3000 cubic centimeters, and 200 times 40 /2 will only greatly increase the total volume to a greater extent if we were to increase the value. If the prism's height remains constant but the triangle that forms its base is twice as tall and twice as wide, the prism's volume is also twice as large. This is due to the fact that the volume of a prism is determined by the area of its base and the prism's height. Because the prism's base has doubled in size while its height has remained constant, the prism's volume has also doubled.
Understanding Volume
The volume of the Toblerone bar that has been provided measures at around 63 cubic centimeters once we multiply the base by height by width, 3.5 x 15 x 2.4. We can assume that 63 cubic centimeters is how much volume it would take to hold 100 grams. Therefore, we can assume that a 126 cubic centimeter bar can hold double 200 grams if we double the amount of volume that it can hold. There are multiply different dimensions that can hold up to 200 grams, including 1, 2, 200 / 1, 4, 100, / 4, 10, 10 / 2, 10, 20 / 4, 5, 20 / , and a ton other factors that can give us a volume of 126 cubic centimeters. Alternatively, we can stretch out the bar and double its length by multiplying 3.5 by 2 to get 7, to get a total dimension of 7, 15, and 2.4 which also gives us a volume of 126 cubic centimeters. Now, how much could hold 360 grams? Well, if we multiply the 126 by 1 / 1/6, we get a total of 147 cubic centimeters which should be capable of holding up to 360 grams. There are numerous different ways which we can reach this number, like multiplying the length of 3.5 by 2 1/3 to get us a number nearly close enough to the value of 147 cubic centimeters
Circles, Diameter, and Cylinders
A circle is a geometric shape made up of all points in a plane that are equidistant, meaning from equal distances, from a fixed point known as the circle's centre. The radius of a circle is the distance from the centre of the circle to any point on the circle. The circumference of a circle is formed by the set of all points in a plane that are equidistant from the centre of a circle. The area of a circle is the region enclosed by a circle's circumference. Circular objects or phenomena, such as wheels, coins, or the planets of the solar system, are frequently represented by circles. Circles usually only has 1 face. Its 3D form, cylinder, but also a sphere- is basically an extended circle. Cylinders typically has 3 faces, 2 vertices or vortexes, and 2 edges- though this may be debated. Many argue about whether or not cylinders have no vortexes and no edges, and some say that they do. This may be due to how it may be curved and not meet the characteristics of it being considered an edge. So, the answer may vary depending on the source- but in my opinion, I think that they have 3 faces, 2 vertices and 2 edges. Please correct me if I'm wrong!
Circles are simple closed curves that are all the same distance from the centre. The circumference of a circle is the distance around the circle's outside and is calculated using the formula C = 2r, where r is the radius of the circle. The area of a circle is the amount of space inside the circle and is calculated using the formula A = r2, where r is the circle's radius. Circles are frequently used to represent symmetry and balance, and they have long been important symbols in many cultures. Geometry is a branch of mathematics that studies circles and other geometric shapes.
Measuring 2D Circles
In order to measure the area within a triangle, we must first know its radius. If you are given a diameter, however, we can simply divide that by 2 to half it to get the radius. If you have a radius and want to get the diameter, the opposite applies- you must multiply the value by 2 to double it. Radius measures the distance from the circles edge to the centre, while diameter measures the distance from the circles edge to the opposite side of the circle. Once we are given the radius, we must multiply it by itself, and multiply it by the value pi, which is the circumference within a circle. Circumference refers to the circles perimeter. Once we have done that, we should have the total area of our circle. For my example, I divided the diameter 10 by half to get radius (5), and multiplied it by itself (5x5) to get 25, and multiplied that value by pi to get the final area. (25 times 3.14 = 78.5).
A cube is a shape where all the sides are equal- length, width, and hight, and where all the corners are a right angle, but, rectangles aren't the same. They do share very similar characteristics, however- like how they both have 6 faces, they both have 8 vertices and 12 edges, making them technically the same shape. But, because rectangles and their edges aren't all the same width and length, rectangles are a special type of cube.
Measuring 3D Circles
Measuring cylinders is quite simple. Gather the information that has been analyzed by measuring a square. The equation is basically grasping a single layer and finding the area, and multiplying it by the amount of layers it has to get the volume. Thats quite the same with circles- and quote what 3D shapes are. They grab one area, and multiply it to gather a series of multiple layers. If we know that the area of our 2D circle is around 78.5, Take the area, and multiply it by the height to get the total volume. Taking the example, we have already identified the area of our original circle. Multiplied by 69 (the height,) the volume equals to roughly 5416. The mathematical formula for a cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder, and where the ^ represents exponents and squared operations.
Sphere
A sphere, like a ball, is a three-dimensional geometric shape that is completely round. A sphere is the three-dimensional equivalent of a circle, consisting of all points in space that are the same distance from a fixed point known as the centre. The radius of a sphere is the distance between its centre and its surface. Because spheres are defined by their radius, two spheres with the same radius are considered to be the same size regardless of where they are in space. V = 4/3r3, where r is the radius of the sphere, is used to calculate the volume of a sphere. Though its not widely discussed yet in grade 8, I think its still cool to take a deeper look into the shape that makes up our world and solar system!
The Coca Cola can has a diameter of 6.62 cm, and a height of 12.2 cm. The juice box has a height of 120 mm, length of 38 mm, and a width of 48 mm.
The Juice + Coca Cola Problem
Before we begin any further, we must first measure the volume of both of these shapes. To measure the volume of a cylinder-like shape, we must find the area, and multiply it by the height of our can of 12.2 centimeters. We were given the diameter, but we must get the radius. This can be done by dividing 6.62 by 2, to get a radius of 3.31. Using exponents, 3.31 multiplied by itself (3.31 times 3.31) gives us a total product of 10.9. We must now multiply that value by the circumference of a circle, aka 3.14 to get the total area of 34.21. Multiply that by the height of our can, 12.2 centimeters, to reach our final volume of 471.8 cubic centimeters.
Now, for the juice box, in order to find the volume, we must multiply base times height times width through 120 x 38 x 48, which equals 218,880 cubic millimeters of the total. Wow, such a big number! But, we must convert that to cubic centimeters in order to match up with how much volume the soda has for the purpose of better referencing. One millimeter is equal to 0.1 centimeters and 1 cm is equal to 10 mm, and therefore, 218,880 cubic millimeters equals to around 218.88 cubic centimeters.
Through this, we can finally compare and contrast the differences in volume between the 2. The soda has a volume of around 471.8 cubic centimeters, while the juice has a volume of 218.88 cubic centimeters- which is a nearly double difference in terms of the volume.
Question ) 1. Approximately how many juice boxes would you need to purchase to have the same amount of drink as one can?
We can represent this question through division since division is seeing how many groups something can fit into something. 471.8 divided by 218.88 is equal to just over 2. This means that in order to have nearly the same volume, we would need to buy 2 juice boxes in order to have nearly the same amount in which we have of soda.
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2. You need 20L of total drink for a party. Determine how many of each type of drink you would need to purchase if you purchase either all cans or all boxes. (1cm 2 = 1mL, 1000mL = 1L)
Soda ) To calculate the amount of soda we need to reach 20L, we must convert 20L to millimeters. 20 liters equals to around 20,000 mL, so, we could just simply divide the 2 values to see how many soda we need. Each can of soda holds around 471.8 cubic centimeters, or We would need around 1,000 cans of soda, each consisting of 471 cubic centimeters. 471 cubic centimeters is equivalent to around 478 milliliter. Dividing 20,000 by 471.8, we would need around 40 total of cans for there to be around 20L of total drinks.
Juice ) To calculate the amount of juice we would need to reach 20L, we have already done most of the work in our last question. 20L is equal to around 20,000mL, so, we could simply just divide the 2 values. 218.88 cubic centimeters is equal to around 218.88 cubic centimeters since 1 cm/3 is equal to 1mL. Then, we can confidently divide 20,000 by 218.88 to get a total of around 91 boxes of juice that's needed in order to meet the criteria of 20 liters. 91 and 40 is nearly a double increase in the needed cans or juice boxes that's needed to sustain the party, and therefore, although the price different only lies with a 10 cents difference, we can confidentaly say that the soda is definitely a better worth of your money.
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3. Your cooler measures 50.3cm x 24.1cm x 29 cm. What is the greatest number of cans that will fit in the cooler? What is the greatest number of juice boxes that will fit? Show your model for each type of drink.
First, let's identify the total volume of the cooler. If we multiply the dimensions of 50.3 times 24.1 times 29, we get a total of 35154 cubic centimeter as the volume of our cooler. We can see how much of each item can fit into our cooler through division. We can do this by dividing the volume of our soda and boxes of juice to see how many times each can fit into the cooler. 35,154 divided by 218.88 shows that around 160 boxes of juice is able to fit into the cooler roughly. In terms of soda, 35,154 divided by 471.8 shows that roughly 74 cans of soda is able to fit into the cooler.
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4. How much does each type of drink cost per L?
We are already given a reference that each can of soda costs 42 cents, and one box of juice costs 33 cents. That's 42 cents for 471.8 cubic centimeters of soda, and 33 cents for 218.88 cubic centimeters worth of soda. The soda costs 42 cents per 471.8 cubic centimeters, then it would cost 1 / (471.8 / 1000) = 1 / 0.4718 = 2.1 times as much per liter. So a liter of this drink would cost 42 cents * 2.1 = $0.88. The juice on the other hand would cost around $1.53 if we multiply the original value of 218.88 by 4.6 to get a liter. Looking at this data alone, I can confidently say that I would rather buy the soda over the juice because although the soda is only 10 cents more, it gives you over double the amount in which the juice gives you, so, the soda overall is a better worth of your money in comparision to the juice.
Calculating Volume from Circumference :
If we are only given the circumference and height of a cylinder, we can use that to identify the circumference. Most of the time, the circumference is 3.14, but, that may change in this scenerio- something that I've just learned is capable of happening. Once we have gathered the circumference and height. We can divide circumference (10 cm/2 in this case) by 3.14 to get the diameter. Here, the diameter is around 3.18. We can divide this by 2 to get 1.59 which is our radius. With this information, we can simply following the normal procedures in finding the volume of a cylinder. 3.18 times 3.18 equals 10.1, multiplied by the values of pi equals 31.7, and times that by the height of 20 to get 634 cubic centimeters.
Calculating Volume from Missing Data :
We are given the base of 30cm, and the volume of 195 cm. We need to identify the height of the cylinder. We can represent this through algebra. 30x=195. If we are trying to see what multiplied what gives us a specific value, we can simply divide. 195 divided by 30 gives us 6.5, meaning the height of the cylinder is 6.5. We can fact check this value by multiplying 30 centimeters, the area of our surface, by the amount of projected layers there are of 6.5. We get a total of 195 centimeters cubed, which is the total volume within our cylinder.
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4. Why do we need to know the height of the cylinder to identify the volume?
Because if we don't have a height and we only identify the base, then technically, we have only identified the area of the 2-dimensional surface. A 3-dimensional shape, aka a cylinder, is identified by its width and combined layers of an ordinary circle, which is why we multiply the area by the height. If we don't multiply by height, then we only have a 2 dimensional figure. To find the volume of a cylinder, we multiply the base area by the height of the cylinder, and if we aren't doing that, we are missing out on half of the equation which gives us a value that will not equate to meeting the requirements within a 3 dimensional figure.
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Cylinders, Triangular Prisms, and Rectangular Prisms
First, lets identify the volume of our cylinder. Assuming it means the area of our cylinder is 16, and the height of 30 centimeters- 30 times 16 equals 420 cubic centimeters as the cylinders volume. That means that the total volume of the contents within the 2 triangular prisms, and rectangular prisms have to equal that exact same amount. First, we can express this through division. 420 divided by 4 equals 105, meaning that if each of the 4 shapes has 105 cubic centimeters in volume, it would work. But, since they have to be different in volume, this cannot work- so, we can try to balance the values on both sides. 105, 105, 105, and 105. For instance, we can subtract 1 from our first value to give us 104, but, we would have to add on the value to another shape. + 1 to our second value of 105 to get us 106 cubic centimeters. We can add on 2 to our third value to get 107, but would need to subtract the value from our fourth value to get 103. At the end, we have a total of : 104, 106, 107, and 103 is the equivalent volumes within our shapes. It doesn't matter what the value may be, they just must have a total sum of 420. Some numbers include 15, 21, 28, 356, 8, 10, 14, 388, 9, 10, 12, 389, etc. Say that we stick with 104, 106, 107, and 103. If we want to find a triangular prism which has a volume that has 104 cubic centimeters, we can multiply it by 2 to get 208, and find 3 factors that equal that amount since the formula for a triangular prism is b x h x w / 2. The volume of our triangular prism may be 1 times 8 times 26 which equals 208, and divided by 2 gets you 104. If we want a triangular prism that gets us 106, we can do the same process. 5 times 6 times 7.0667 gives us roughly an answer close to 106. If we want a rectangular prism that equals 107 and 103, we can just find 3 factors that equal those amounts to get base times height times width. An instance is 1 times 2 times 53.5, and 1 times 2 times 51.5.
