The surface area of a 3D shape is the sum of the areas of each face. Surface area is also measured in square units such as square centimeters (\(\text{cm}^2\) ), square inches ( \(\text{in}^2\) ) 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 side length \(s\) is calculated as:
\(SA = 6 s^2\)
Cuboid
The surface area of a cuboid (rectangular prism) with length \(l\), width \(w\) and height \(h\) is calculated as:
\(SA = 2 (l w + l h + w h)\)
Cone
The surface area of a cone with height \(h\), slant \(l\), and radius \(r\) is calculated as:
\(SA = \pi r (l + r)\)
Cylinder
The surface area of a cylinder with radius \(r\) and height \(h\) is calculated as:
\(SA = 2 \pi r (h + r)\)
Triangular Prism
The surface area of a triangular prism with two equal side lengths \(s\), base \(b\), height \(h\) and length \(l\) is calculated as:
\(SA = bh + 2ls + lb\)
Sphere
The surface area of a sphere with radius \(r\) is calculated as:
\(SA = 4 \pi r^2 \)
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 prism 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.
\(SA_{\text{prism}} = (2)(6) + (3)(6) + (2)(6) + (3)(6) + (3)(2) + (3)(2) \)
\(SA_{\text{prism}} = 2 ( (2)(6) + (3)(6) + (3)(2) )\)
\(SA_{\text{prism}} = 2 (12 + 18 + 6)\)
\(SA_{\text{prism}} = 72 \; [\text{cm}^2]\)
Therefore, we can determine that the surface area of the prism is \(72 \; [\text{cm}^2]\) .
A sphere has a radius \(6\) inches, what is its surface area?
Show Answer
The surface area of a sphere can be calculated by using its corresponding formula:
\(SA_{\text{sphere}} = 4 \pi r^2 \)
\(SA_{\text{sphere}} = 4 \pi (6)^2 \)
\(SA_{\text{sphere}} = 452.39 \; [\text{in}^2]\)
Therefore, we can determine that the surface area of the sphere is \(452.39 \; [\text{in}^2]\).
Juan is making a yard sign in the shape of a rectangular prism. The yard sign will be \(60\;[\text{cm}]\) long, \(40\;[\text{cm}]\) high and \(5\;[\text{cm}]\) thick. The cost of painting the yard sign is \($0.002\) per square \(\text{cm}\). How much does Juan have to spend to paint the yard sign?
Show Answer
First, we can calculate the surface area of the sign. We can do so by adding the area of all \(6\) sides of a rectangular prism:
\(SA_{\text{prism}} = 2 (l w + l h + w h)\)
We can now plug in the appropriate values and simplify to determine the surface area:
\(SA_{\text{sign}} = 2 ((60)(5) + (60)(40) + (5)(40)) \)
\(SA_{\text{sign}} = 2 (300 + 2400 + 200)\)
\(SA_{\text{sign}} = 5800 \; [\text{cm}^2]\)
Next, we can calculate the cost of painting by multiplying the rate by the surface area since Juan will need to paint all sides of the sign.
\(\text{Cost} = SA \cdot \text{rate}\)
\(\text{Cost} = (5800 \; [\text{cm}^2])\left(\cfrac{$0.002}{[\text{cm}^2]}\right)\)
\(\text{Cost} = $11.60\)
Therefore, we can determine that it will cost Juan \($11.60\) to have the yard sign painted.
Draw a net for the shape below and calculate the surface area.
Show Answer
First, we can draw the net. The top and bottom of the cylinder are circles. The middle portion will unfold to a rectangle. The width of the rectangle is the circumference of the circle! Recall the circumference of a circle is \(\pi d = 2 \pi r\).
The surface area is \(2\) circles plus a rectangle:
\(SA = 2 \pi r^2 + 2 \pi r h\)
We can use the formula below, which is a simplifed version, to determine the surface area of the cylinder:
\(SA_{\text{cylinder}} = 2 \pi r (h + r)\)
\(SA_{\text{cylinder}} = 2 \pi \cfrac{4}{2} (10 + \cfrac{4}{2})\)
\(SA_{\text{cylinder}} = 4 \pi (12)\)
\(SA_{\text{cylinder}} = 150.80 \; [\text{cm}^2]\)
Therefore, we can determine that the surface area of the cylinder is \(150.80 \; [\text{cm}^2]\).