1. Surface Area of 3 – Dimensional Figures
Cubes and Rectangular Prisms

2. Definition
Surface area is the sum of the areas of all the surfaces of a 3 – Dimensional figure.
It is basically the outside layer or surface.
Example: When you paint a wall, you are painting the surface of the wall.

3. Surface Area of Cubes
Remember that a cube is made up of squares on all sides. There are 6 sides to a cube, just like dice. Therefore, there are 6 squares total. To find the surface area, find the area of one square and then multiply by 6. Simply use the following formula.
S.A. = 6s2

4. Example Find the surface area of the cube.
4 cm
4 cm
Find the surface area of the cube.
Recall that all of the sides of a square are the same length. So, all you have to do is multiply the sides.
Find the area first. The length and the width are both 4 cm. 4 x 4 = 16. Thus, the area of one square is 16 cm2. Now multiply by 6 and we see that the surface area is 96 cm2.

5. YOUR TURN!!! Find the surface area of the following cube.
7 mm
7 mm
Find the surface area of the following cube.
What is the length and width of the cube? 7 mm
Now, find the area of one square. 7 x 7 = 49. Thus, the area of one square is 49 mm2.
Take this and multiply by 6 and 49 x 6 =
294 mm2.
So, the surface area of the cube is 294 mm2.

6. Rectangular Prisms
To find the surface area of a prism, you have to find the areas of all the sides and add them up.
Understand that there are still 6 sides to a prism and that opposite sides are the same shape.
To find the surface area of a rectangular prism use the following formula.
S.A. = 2(wh + lh + lw)

7. Example Find the surface area of the prism.
S.A. = 2(1 x x x 1)
S.A. = 2 ( )
S.A. = 2 (103)
S.A. = 206
The surface area is 206 m2. Surface area is just like finding the regular area. The unit of measurement is squared.

8. TAKE THE CHALLENGE! You find the surface area of the following prism.

9. CHALLENGE Cont. S.A. = 2(4 x 7 + 26 x 7 + 26 x 4)
The surface area is 628 ft2

10. SAMPLE ARMT QUESTION
David has two shapes he is painting for a project. For each of the two shapes, he will paint only the outside including the lid.
One shape is a cylindrical can with a radius of 4 inches and a height of 9 inches.
The other shape is a rectangular prism-shaped box. The box is 3 inches wide, 8 inches long, and 5 inches high.
Which shape has the greater surface area for David to paint?

11. Sample ARMT Question Cont.
First, find the surface area of the cylinder.
Formula: (2 x 3.14 x r x h) + (2 x 3.14 x r2)
Stick the numbers in directly.
(2 x 3.14 x 4 x 9) + (2 x 3.14 x 4 x 4)
This equals
Add these together and the surface area of the cylinder is square inches.

12. Sample ARMT Question Cont.
Next, find the surface area of the rectangular prism.
Formula: 2(wh + lh + lw)
Stick the numbers in directly.
2(3·5 + 8·5 + 8·3)
This equals 2( )
This equals 2(79) = 158
The surface area of the rectangular prism is 158 square inches.

13. ANSWER The cylinder has a surface area of 326.56 square inches.
The rectangular prism has a surface area of 158 square inches.
The cylindrical can has the greater surface area to paint.