Surface Area of Prisms

The surface area of a 3D figure is the total area of all its outside faces. To find the surface area of a prism, you simply calculate the area of each flat face and add them all together.

Using a Net

A great way to visualize surface area is by using a net. A net is a 2D drawing of all the faces of a 3D shape laid out flat. If you fold up a net, it creates the 3D shape. Drawing a net helps ensure you don't forget to calculate the area of any face!

Rectangular Prisms

A rectangular prism has $6$ rectangular faces. These faces come in three identical pairs: top and bottom, front and back, and left and right.

The formula for the surface area ($SA$) of a rectangular prism with length $l$, width $w$, and height $h$ is:

$SA = 2lw + 2lh + 2wh$

Example 1: Basic Rectangular Prism

Find the surface area of a rectangular prism with dimensions $4 \times 3 \times 2$.

1. Find the area of the top and bottom: $2 \times (4 \times 3) = 2 \times 12 = 24$
2. Find the area of the front and back: $2 \times (4 \times 2) = 2 \times 8 = 16$
3. Find the area of the left and right sides: $2 \times (3 \times 2) = 2 \times 6 = 12$
4. Add them together: $24 + 16 + 12 = 52$

The total surface area is $52$ square units.

Example 2: The Wrapping Paper Problem

How much wrapping paper is needed to completely cover a box that is $10 \times 8 \times 5$ inches?

This is a real-world surface area problem!

$SA = 2(10 \times 8) + 2(10 \times 5) + 2(8 \times 5)$ $SA = 2(80) + 2(50) + 2(40)$ $SA = 160 + 100 + 80 = 340$

You would need $340$ square inches of wrapping paper.

Triangular Prisms

A triangular prism has $5$ faces in total: $2$ identical triangular bases and $3$ rectangular sides.

To find the surface area:

1. Find the area of the two triangular bases using the formula $A = \frac{1}{2}bh$.
2. Find the area of the three rectangular sides (length $\times$ width).
3. Add all five areas together.

Example 3: Triangular Prism

Find the surface area of a triangular prism where the triangular bases have a base of $6$ cm, a height of $4$ cm, and side lengths of $5$ cm. The length of the prism is $10$ cm.

1. Triangular bases: Area of one triangle = $\frac{1}{2} \times 6 \times 4 = 12$ cm$^2$. Since there are two identical triangles, their combined area is $12 + 12 = 24$ cm$^2$.

2. Rectangular sides: The three rectangles connect to the three sides of the triangle ($6$ cm, $5$ cm, and $5$ cm) and are all $10$ cm long.

   • Bottom rectangle: $6 \times 10 = 60$ cm$^2$
   • Side rectangle 1: $5 \times 10 = 50$ cm$^2$
   • Side rectangle 2: $5 \times 10 = 50$ cm$^2$

3. Total Surface Area: $24 + 60 + 50 + 50 = 184$

The total surface area is $184$ cm$^2$.