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Surface Area of Solids Practice Questions

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Updated on Oct 3, 2011

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

Surface Area of Solids Practice Questions

Practice

  1. A rectangular solid measures 4 units by 5 units by 6 units. What is the surface area of the solid?
  2. Danielle's cube has a volume of 512 in3. What is the surface area of her cube?
  3. The surface area of a rectangular solid is 192 cm2. 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?
  4. The volume of a cube is x3 cubic units and the surface area of the cube is x3 square units. What is the value of x?
  5. 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

  1. 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.
  2. 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.
  3. 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 + 8w + 24w = 96 + 32w. Because the surface area of the solid is 192 cm2, 96 + 32w =192, 32w = 96, w = 3. The width of the solid is 3 units.
  4. 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 x3,which is equal to x, because (x)(x)(x) = x3. 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) =x2. A cube has six aces, so the total surface area of the cube is equal to 6x2 square units. It is given that the surface area of the square is x3 square units. Therefore, 6x2 = x3. Divide both sides by x2, and the value of x is 6.
  5. 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 2x is the height of the solid and 4x is the width of the solid. Two faces of the solid measure x units by 2x units, two faces measure x units by 4x units, and two faces measure 2x units by 4x units. Therefore, the surface area of the solid is equal to 2(x × 2x) + 2(x × 4x) + 2(2x × 4x) = 2(2x2) +2(4x2) + 2(8x2) = 4x2 + 8x2 + 16x2 = 28x2.

 

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