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

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 the formula:

\(SA = 4 \pi r^2 \)

\(SA = 4 \pi (6)^2 \)

\(SA = 452.39 \; \text{in}^2\)

The surface area 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 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?

Show Answer
Start by calculating the surface area of the sign.

The surface area of a rectangular prism is calculated by adding the area of all 6 sides of prism.

\(SA = 2 (l w + l h + w h)\)

\(SA = 2 ((60)(5) + (60)(40) + (5)(40)) \)

\(SA = 2 (300 + 2400 + 200)\)

\(SA = 5800 \; \text{cm}^2\)

Next, we calcualte the cost of painting by multiplying the rate and the sufrace area since Juan will need to paint all sides of the sign.

\( C = 5800 \; \text{cm}^2 \cdot \cfrac{$0.002}{\text{cm}^2}\)

\( C = $11.60\)

The cost to paint the sign is \( $11.60\).

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

Show Answer
First draw the net. The top and bottom of the cylinder are circles. The middle portion will unfold to a rectangle. The width of the rectanle is the circiumference of the circle!
Recall the circiumference 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:

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

\(SA = 2 \pi \cfrac{4}{2} (10 + \cfrac{4}{2})\)

\(SA = 4 \pi (12)\)

\(SA = 150.80 \; \text{cm}^2\)

The surface area is \( 150.80 \; \text{cm}^2\).