Surface Area
The surface area of a 3D shape is the sum of the areas of each face. Sureace area is also measured in square units such as square centimeters ( \(cm^2\) ), square inches ( \(in^2\) ) etc..
Surface Area of Common Shapes
Let's look at how to calculate the surface area of the shapes below:
Cube
The surface area of a cube with length s is calculated by:
\(SA = 6 s^2\)
Cuboid
The surface area of a cuboid (rectangular prism) with length l, width w and height h is calculated by:
\(SA = 2 (l w + l h + w h)\)
Cone
The surface area of a cone with height h, slant l and radius r is calculated by:
\(SA = \pi r (l + r)\)
Cylinder
The surface area of a cylinder with radius r and height h is calculated by:
\(SA = 2 \pi r (h + r)\)
Triangular Prism
The surface area of a triangular prism with two equal sides s, base b, height h and length l is calculated by:
\(SA = bh + 2ls + lb\)
Sphere
The surface area of a sphere with radius r is calculated by:
\(SA = 4 \pi r^2 \)
Surface Area Using Nets
The net of an 3D shape is formed when the shape is unfolded along its edges and its faces are laid out in a pattern in 2D. Nets are helpful to visualize the different faces of a 3D object to calculate the surface area.
Let's draw a net for the shape below.
We have to unfold the 3D object to get the net shape below:
You can see that the original 3D shape is made up of 6 rectangles. We can calculate the area of each using \(A = lw\). Lastly, we add them all up to get the surface area.
\(SA = (2)(6) + (3)(6) + (2)(6) + (3)(6) + (3)(2) + (3)(2) \)
\(SA = 2 ( (2)(6) + (3)(6) + (3)(2) )\)
\(SA = 2 (12 + 18 + 6)\)
\(SA = 72 \; \text{cm}^2\)
The surface area is \(72 \; \text{cm}^2\) .
