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MODULE 6 - AREAS OF PLANE FIGURES
OVERVIEW
Plane geometry is the study of geometric figures that can be drawn on a two-dimensional surface called plane. (Earnhart, 2011) This module deals with the common plane figures such as triangles, quadrilaterals, circles, and polygons including their properties, relations and measurement.
Plane geometry is the study of geometric figures that can be drawn on a two-dimensional surface called plane. (Earnhart, 2011) This module deals with the common plane figures such as triangles, quadrilaterals, circles, and polygons including their properties, relations and measurement.
LEARNING OBJECTIVES:
At the end of this module, you should be able to:1. Identify the properties of a plane figure2. Indicate the formula for the given plane figure.3. Calculate the area of the given plane figure.
At the end of this module, you should be able to:1. Identify the properties of a plane figure2. Indicate the formula for the given plane figure.3. Calculate the area of the given plane figure.
AREA OF A TRIANGLE
In a two-dimensional plane, the area of a triangle is defined as the total space occupied by its three sides. The area of a triangle is equal to half the product of its base and height, therefore A = 1/2bh is the fundamental formula. This formula works for any triangle, whether it's a scalene triangle, an isosceles triangle, or an equilateral triangle. It's important to remember that a triangle's base and height are perpendicular to each other. However, various formulas can also be used in computing for the area of a triangle such as the side-angle-side (SAS) formula and the Heron's formula. Shown at the left is the table of formulas.
In a two-dimensional plane, the area of a triangle is defined as the total space occupied by its three sides. The area of a triangle is equal to half the product of its base and height, therefore A = 1/2bh is the fundamental formula. This formula works for any triangle, whether it's a scalene triangle, an isosceles triangle, or an equilateral triangle. It's important to remember that a triangle's base and height are perpendicular to each other. However, various formulas can also be used in computing for the area of a triangle such as the side-angle-side (SAS) formula and the Heron's formula. Shown at the left is the table of formulas.
As an example, I got the figure on the right from my activity 5. The height and base of the triangular ruler were already given so it would be easy to use the general formula stated above. One-half times eight times four is 16. Thus, the area of the triangular ruler is 16 square inches.
However, in case the given for a triangle are two sides and the included angle, it is advisable to use the Side-Angle-Side (SAS) formula for convenience. In the same manner, in case the three sides are given, then use the Heron's formula.
However, in case the given for a triangle are two sides and the included angle, it is advisable to use the Side-Angle-Side (SAS) formula for convenience. In the same manner, in case the three sides are given, then use the Heron's formula.
QUADRILATERALS
A quadrilateral is a general term for a polygon with four sides and four vertices. Both triangles and quadrilaterals include sides, vertex, interior angles, height and base except that a quadrilateral has a diagonal. Here are the properties of a quadrilateral:
A quadrilateral is a general term for a polygon with four sides and four vertices. Both triangles and quadrilaterals include sides, vertex, interior angles, height and base except that a quadrilateral has a diagonal. Here are the properties of a quadrilateral:
- They have four sides.
- They have four vertices.
- They have two diagonals.
- The sum of all interior angles is 360°.
RECTANGLE
A rectangle is a type of quadrilateral that has its parallel sides equal to each other and all the four vertices are equal to 90 degrees. Hence, it is also called an equiangular quadrilateral. Since, the opposite sides are equal and parallel, in rectangle, therefore, it can also be termed as a parallelogram. While a rectangle is a type of quadrilateral, parallelogram, closed shape and plane figure, only a square is always a type of rectangle. The area of a rectangle is just base times height or length times width (A = bh).Shown on the right is from my activity 5. As you can see, to get the area of the rectangular picture frame, just multiply the base and height of the rectangle. For instance, 14 times 11 is equal to 154. Thus, the area is 154 square inches.
A rectangle is a type of quadrilateral that has its parallel sides equal to each other and all the four vertices are equal to 90 degrees. Hence, it is also called an equiangular quadrilateral. Since, the opposite sides are equal and parallel, in rectangle, therefore, it can also be termed as a parallelogram. While a rectangle is a type of quadrilateral, parallelogram, closed shape and plane figure, only a square is always a type of rectangle. The area of a rectangle is just base times height or length times width (A = bh).Shown on the right is from my activity 5. As you can see, to get the area of the rectangular picture frame, just multiply the base and height of the rectangle. For instance, 14 times 11 is equal to 154. Thus, the area is 154 square inches.
SQUARE
Square is a type of quadrilateral that has four parallel sides with equal lengths and equal interior angles which is 90 degrees. Further, a square is a type of parallelogram, rectangle, and rhombus. It is a parallelogram because it has two pairs of parallel, congruent sides. It is a rectangle because it has two pairs of parallel, congruent sides with four congruent interior angles. It is a rhombus because it has four congruent sides. The area of a square is just the square of its side (A = s2).
On the left is an example from activity 5. To get the area of the mosquito box that is square in shape, just multiply its side by itself or square the side. For instance, 52 is equal to 25, thus, the area is 25 square inches.
Square is a type of quadrilateral that has four parallel sides with equal lengths and equal interior angles which is 90 degrees. Further, a square is a type of parallelogram, rectangle, and rhombus. It is a parallelogram because it has two pairs of parallel, congruent sides. It is a rectangle because it has two pairs of parallel, congruent sides with four congruent interior angles. It is a rhombus because it has four congruent sides. The area of a square is just the square of its side (A = s2).
On the left is an example from activity 5. To get the area of the mosquito box that is square in shape, just multiply its side by itself or square the side. For instance, 52 is equal to 25, thus, the area is 25 square inches.
RHOMBUS
A rhombus is a quadrilateral (plane figure, closed shape, four sides) with four equal-length sides and opposite sides parallel to each other. All rhombuses are parallelograms, but not all parallelograms are rhombuses. All squares are rhombuses, but not all rhombuses are squares. The opposite interior angles of rhombuses are congruent. Diagonals of a rhombus always bisect each other at right angles.
In computing for the area of a rhombus, there are three formulas given in the figure. First is 1/2ab or the half of the product of the two diagonals. The second is sh or bh or the product of one side and the altitude or the height. The last formula can be used if an angle is given. It will just be A = s2 sin θ.
A rhombus is a quadrilateral (plane figure, closed shape, four sides) with four equal-length sides and opposite sides parallel to each other. All rhombuses are parallelograms, but not all parallelograms are rhombuses. All squares are rhombuses, but not all rhombuses are squares. The opposite interior angles of rhombuses are congruent. Diagonals of a rhombus always bisect each other at right angles.
In computing for the area of a rhombus, there are three formulas given in the figure. First is 1/2ab or the half of the product of the two diagonals. The second is sh or bh or the product of one side and the altitude or the height. The last formula can be used if an angle is given. It will just be A = s2 sin θ.
PARALLELOGRAM
A parallelogram is a flat shape with four straight, connected sides so that opposite sides are congruent and parallel. This means a parallelogram is a plane figure, a closed shape, and a quadrilateral. Other shapes are types of parallelograms. Rhombus is a parallelogram with four congruent sides. Rectangle is a parallelogram with four congruent interior angles. Square is a parallelogram with four congruent sides and four congruent interior angles.
Finding the area of a parallelogram is simply multiplying the base and the height just like in rectabgle and rhombus. Area of parallelogram = bh.
A parallelogram is a flat shape with four straight, connected sides so that opposite sides are congruent and parallel. This means a parallelogram is a plane figure, a closed shape, and a quadrilateral. Other shapes are types of parallelograms. Rhombus is a parallelogram with four congruent sides. Rectangle is a parallelogram with four congruent interior angles. Square is a parallelogram with four congruent sides and four congruent interior angles.
Finding the area of a parallelogram is simply multiplying the base and the height just like in rectabgle and rhombus. Area of parallelogram = bh.
TRAPEZOID
A trapezoid is a quadrilateral with one pair of parallel sides. A trapezoid is a plane figure (flat), a closed figure (it has an interior and exterior), a polygon (straight sides), and a quadrilateral (four straight sides). Note that an isosceles trapezoid is a trapezoid whose non-parallel sides are equal in length. The area of a trapezoid is the average of the bases multiplied by the altitude or height [A = 1/2(a + b)h].
On the right is taken from my activity 5. The task was easy because the height and bases were already given. I just substituted the values into the formula and I got 56.25 square inches as the area of the trapezoidal dustpan.
POLYGONS
A polygon is a closed plane figure with at least three straight sides and angles. Both triangles and quadrilaterals are an example of a polygon, in particular a regular polygon. Regular polygons are polygons with equal sides and included angles.
The figure on the left shows the regular polygons from triangle with 3 sides to dodecagon with 12 sides.
A polygon is a closed plane figure with at least three straight sides and angles. Both triangles and quadrilaterals are an example of a polygon, in particular a regular polygon. Regular polygons are polygons with equal sides and included angles.
The figure on the left shows the regular polygons from triangle with 3 sides to dodecagon with 12 sides.
FINDING THE AREA OF A REGULAR POLYGON
Regular polygons have all straight sides equal in length and all interior angles equal. Finding the area of any regular polygon (the space of the interior) is easy if you know what an apothem is. Apothem is basically the height of the triangle formed from the regular polygon. It is also the distance of a perpendicular line from any side of the polygon to its center. On the right is the formula in finding the area of any regular polygon. In some cases that the length of the apothem is not given, we can always find it through tangent especially that one of the angles of a triangle in any polygon can be easily derived, just divide 360 degrees to the number sides of the polygon.
Regular polygons have all straight sides equal in length and all interior angles equal. Finding the area of any regular polygon (the space of the interior) is easy if you know what an apothem is. Apothem is basically the height of the triangle formed from the regular polygon. It is also the distance of a perpendicular line from any side of the polygon to its center. On the right is the formula in finding the area of any regular polygon. In some cases that the length of the apothem is not given, we can always find it through tangent especially that one of the angles of a triangle in any polygon can be easily derived, just divide 360 degrees to the number sides of the polygon.
CIRCLES
Circles is a locus of points equidistant to a fixed point. The constant distance is the radius and the fixed point is the center. A line segment connecting two points on the circle and going through the center is called a diameter of the circle. The distance around the circle is called the circumference and is equal to 2πr.
On the left is a table of circle formulas from a circle, semicircle, quadrant, ring, sector, to segment. Aside from area, perimeter is also included. Indeed, the area of a circle is equal to πr2. Note that a semicircle is half of the circle; a quadrant is one-fourth of the circle; a ring of a circle or also known as annulus is the region between two concentric circles; a sector is the figure bounded by two radii and an included arc; and, a segment is the region divided by the chord.
Circles is a locus of points equidistant to a fixed point. The constant distance is the radius and the fixed point is the center. A line segment connecting two points on the circle and going through the center is called a diameter of the circle. The distance around the circle is called the circumference and is equal to 2πr.
On the left is a table of circle formulas from a circle, semicircle, quadrant, ring, sector, to segment. Aside from area, perimeter is also included. Indeed, the area of a circle is equal to πr2. Note that a semicircle is half of the circle; a quadrant is one-fourth of the circle; a ring of a circle or also known as annulus is the region between two concentric circles; a sector is the figure bounded by two radii and an included arc; and, a segment is the region divided by the chord.
Example from my activity 5: An electric fan (that is usually circular in shape) has a radius of 8 inches. Find its area.
Simply, we will just use the formula which is A = πr2 and substitute the 8 inches in the r. For instance, 82 is equal to 64 and 64 times π is 201.06. Henceforth, the area of the electric fan is approximately 201 square inches.
Simply, we will just use the formula which is A = πr2 and substitute the 8 inches in the r. For instance, 82 is equal to 64 and 64 times π is 201.06. Henceforth, the area of the electric fan is approximately 201 square inches.
Salazar-Activity5-BSCE1B.pdfACTIVITY #5 (100/100)
REFLECTION
In this module, I was able to relearn the formulas that were already taught to us since elementary. The areas of plane figures was a bit easy for me since it served as a review. Among the topics, I really focused on the area of a regular polygon. I learned that a regular polygon can be broken into triangles through its diagonals. Then, to get the area, we simply need to find the area of the triangle and multiply it to the number of triangles formed or just the number of sides. In finding the area of a triangle, we can either use the general formula which is 1/2ab or use the apothem introduced above. It is up to us which formula to use. What's important is we can solve for the area of the regular polygon in any way we want and we get the area of the polygon precisely and correctly. There is also a general formula aforementioned above to find the area of a regular polygon, but again it is up to us which to use as long we get the correct area of the polygon. Further, our teacher was able to discussed the topics understandable by everybody even with a limited amount of time.
In this module, I was able to relearn the formulas that were already taught to us since elementary. The areas of plane figures was a bit easy for me since it served as a review. Among the topics, I really focused on the area of a regular polygon. I learned that a regular polygon can be broken into triangles through its diagonals. Then, to get the area, we simply need to find the area of the triangle and multiply it to the number of triangles formed or just the number of sides. In finding the area of a triangle, we can either use the general formula which is 1/2ab or use the apothem introduced above. It is up to us which formula to use. What's important is we can solve for the area of the regular polygon in any way we want and we get the area of the polygon precisely and correctly. There is also a general formula aforementioned above to find the area of a regular polygon, but again it is up to us which to use as long we get the correct area of the polygon. Further, our teacher was able to discussed the topics understandable by everybody even with a limited amount of time.
REFERENCES:
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