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

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The surface area of a cube with length s is calculated by:

\(SA = 6 s^2\)

Cuboid

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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

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The surface area of a cone with height h, slant l and radius r is calculated by:

\(SA = \pi r (l + r)\)

Cylinder

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The surface area of a cylinder with radius r and height h is calculated by:

\(SA = 2 \pi r (h + r)\)

Triangular Prism

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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

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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.

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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 \(72 \; [\text{cm}^2]\) .


A sphere has a radius \(6\) inches, what is its surface area?

Juan is making a yard sign in the shape of a rectangular prism. The yard sign will be \(60\;[cm]\) long, \(40\;[cm]\) high and \(5 \;[cm]\) thick. The cost of painting the yard sign is $0.002 per square cm. How much does Juan have to spend to paint the yard sign?

Draw a net for the shape below and calculate the surface area.
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