By LearningExpress Editors

Updated on Oct 3, 2011

To review these concepts, go to Surface Area of Solids Study Guide.

**Surface Area of Solids Practice Questions**

**Practice**

- A rectangular solid measures 4 units by 5 units by 6 units. What is the surface area of the solid?
- Danielle's cube has a volume of 512 in
^{3}. What is the surface area of her cube? - The surface area of a rectangular solid is 192 cm
^{2}. If the height of the solid is 4 units and the length of the solid is 12 units, what is the width of the solid? - The volume of a cube is
*x*^{3}cubic units and the surface area of the cube is*x*^{3}square units. What is the value of*x*? - The width of a rectangular solid is twice the height of the solid, and the height of the solid is twice the length of the solid. If
*x*is the length of the solid, what is the surface area of the solid in terms of*x*?

**Solutions**

- The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has six rectangular faces. Two faces measure 4 units by 5 units, two faces measure 4 units by 6 units, and two faces measure 5 units by 6 units. Therefore, the surface area of the solid is equal to 2(4 × 5) + 2(4 × 6) + 2(5 × 6) = 2(20) + 2(24) + 2(30) = 40 + 48 + 60 =148 square units.
- The volume of a cube is equal to the product of its length, width, and height. The length, width, and height of a cube are identical in measure, so the measure of one edge of Danielle's cube is equal to the cube root of 512, which is equal to 8, because (8)(8)(8) = 512. The area of one face of the cube is equal to the product of the length and width of that face. Because every length and width of the cube is 8 units, the area of one face of the cube is (8)(8) = 64 square units. A cube has six faces, so the total surface area of the cube is equal to (64)(6) = 384 square units.
- The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has six rectangular faces. If
*w*is the width of the solid, then two faces measure 4 units by 12 units, two faces measure 4 units by*w*units, and two faces measure 12 units by*w*units. Therefore, the surface area of the solid is equal to 2(4 × 12) + 2(4 ×*w*) + 2(12 ×*w*) = 96 + 8*w*+ 24*w*= 96 + 32*w*. Because the surface area of the solid is 192 cm^{2}, 96 + 32*w*=192, 32*w*= 96,*w*= 3. The width of the solid is 3 units. - The volume of a cube is equal to the product of its length, width, and height. The length, width, and height of a cube are identical in measure, so the measure of one edge of the cube is equal to the cube root of
*x*^{3},which is equal to*x*, because (*x*)(*x*)(*x*) =*x*^{3}. The area of one face of the cube is equal to the product of the length and width of that ace. Because every length and width of the cube is*x*, the area of any one face of the cube is (*x*)(*x*) =*x*^{2}. A cube has six aces, so the total surface area of the cube is equal to 6*x*^{2}square units. It is given that the surface area of the square is*x*^{3}square units. Therefore, 6*x*^{2}=*x*^{3}. Divide both sides by*x*^{2}, and the value of*x*is 6. - The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has six rectangular faces. If
*x*is the length of the solid, then 2*x*is the height of the solid and 4*x*is the width of the solid. Two faces of the solid measure*x*units by 2*x*units, two faces measure*x*units by 4*x*units, and two faces measure 2*x*units by 4*x*units. Therefore, the surface area of the solid is equal to 2(*x*× 2*x*) + 2(*x*× 4*x*) + 2(2*x*× 4*x*) = 2(2*x*^{2}) +2(4*x*^{2}) + 2(8*x*^{2}) = 4*x*^{2}+ 8*x*^{2}+ 16*x*^{2}= 28*x*^{2}.

From Basic Math in 15 Minutes A Day. Copyright © 2008 by LearningExpress, LLC. All Rights Reserved.

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