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**Surface Area of 3 – Dimensional Figures**

Cubes and Rectangular Prisms

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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.

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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

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**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.

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**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.

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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)

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**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.

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

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**CHALLENGE Cont. S.A. = 2(4 x 7 + 26 x 7 + 26 x 4)**

The surface area is 628 ft2

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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?

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**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.

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**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.

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**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.

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