Theory
Surfaces
To calculate the area of a 2D shape we usually reduce it into three cases: circles, rectangles and triangle. So we only need to remember the formulas for those two shapes.
Triangle
Rectangle
Circle
3D shapes
A three-dimensional shape is a solid shape that has height and depth. For example, a sphere and a cube are three-dimensional, but a circle and a square are not.
A prism is a three-dimensional shape that has non-curved sides. A cube is a prism, but a sphere is not. A prism has a pair of congruent sides, called bases, like the cube, triangular prism and the rectangular prism. Don't confuse a prism with a pyramid, which only has one base.
When we are finding the surface area of a 3-D shape, think of it as unfolding the shape, or flattening it out, and then finding the area of each side. When we add all of these areas up, we have the surface area. There are two types of 3-D shapes we will need to find the surface area of - prisms and non-prisms.
When we are looking for the surface area of a prism, we add all of the areas together to find the total. Another way to find the area of a prism is to find the perimeter of the base and multiply by the height.
SA (of a prism) = (perimeter of the base) * h + (area of the bases)
In order to find the area of a 3-D shape, we must know how to find the area of the basic shapes that make up the sides of the 3-D shape. Here is a list of basic shape formulas to help with finding the surface area of the 3-D shapes:
Cuboid
In cuboids, faces are rectangles.
They have 6 faces, 12 edges and 8 vertices
The area comes by:
A = 2ab + 2ac + 2bc =
= 2(ab + ac + bc)
The volume is
V = a ⋅b ⋅ c
Prisms
A polyhedron with two congruent, parallel bases that are polygons, and all remaining faces parallelograms.
The area comes by:
A = 2 · area of base + area of faces
The volume is
V = area of the base · heigh
Pyramids
A pyramid is like a prism but when all the lateral faces are joined in one point called the apex of the pyramid.
The area comes by:
A = area of base + area of triangular faces
The volume is
V = 1/3 · area of the base · heigh
Cylinders
Cylinders are like prisms, but their cross-section is a circle.
The area comes by:
A = 2 𝛑 R2 + 2𝛑R h
The volume is
V = 𝛑 R2 h
Cones
A cone is a solid bounded by a curved surface that has a common point (vertex or apex) and a circle as the base of the cone.
Slant height of the cone is the straight line that joins the vertex with the circle of the base (g).
The area comes by:
A = 𝛑 R2 + 𝛑 R g
The volume is
V =1/3 𝛑 R2 h
Sphere
In a sphere all points are at the same distance r from the centre of the sphere C.
The distance from the centre to the surface of the sphere is called the radius of the sphere.
The area comes by:
A = 4 𝛑 R2 (4 times the maximun circle of it)
The volume is
V =4/3 𝛑 R3
