GMAT Math : Calculating the surface area of a prism

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Example Question #11 : Prisms

The length of a rectangular prism is twice its width and five times its height. If is the width, give the surface area in terms of .

Possible Answers:

$16W^{2}$

$\frac{13}{5}W^{2}$

$\frac{32}{5}W^{2}$

$\frac{13}{10}W^{2}$

$\frac{16}{5}W^{2}$

Correct answer:

$\frac{32}{5}W^{2}$

Explanation:

If we let be the width, then, since the length is twice this,

L = 2W .

Since the length is five times the height, the height is one fifth the length, so $H = \frac{1}{5}L = \frac{1}{5} \cdot 2W = \frac{2}{5} W$ .

The surface area of a rectangular solid is

S = 2LH + 2LW + 2WH .

which can be rewritten as

$S = 2 \cdot 2W \cdot \frac{2}{5} W + 2 \cdot 2W \cdot W + 2W \cdot \frac{2}{5} W$

$S = \frac{8}{5} W^{2} + 4W^{2} + \frac{4}{5} W^{2} = \frac{32}{5}W^{2}$

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Example Question #61 : Rectangular Solids & Cylinders

Each base of a right prism is a regular hexagon with sidelength 6. Its height is two thirds the perimeter of a base. Give the surface area of the prism.

Possible Answers:

$864+ 108\sqrt{3}$

972

$972\sqrt{3}$

$1,296\sqrt{3}$

$864+ 54\sqrt{3}$

Correct answer:

$864+ 108\sqrt{3}$

Explanation:

The perimeter of a regular hexagon with sidelength 6 is

$P= 6 \cdot 6 = 36$

The height of the prism is two thirds of this, so

$h = \frac{2}{3} \cdot 36 = 24$ .

The lateral area of the prism is the product of the perimeter of a base and the height of the prism, so

$L =Ph= 36 \cdot 24 = 864$

The area of each base can be calculated using the area formula for a regular hexagon:

$B = \frac{3s^{2}\sqrt{3}}{2}$

$B = \frac{3\cdot 6^{2}\sqrt{3}}{2}$

$B = \frac{108\sqrt{3}}{2}$

$B = 54\sqrt{3}$

The surface area is the sum of the lateral area and the areas of the bases:

$S=L+B+B= 864+ 54\sqrt{3} + 54\sqrt{3} = 864+ 108\sqrt{3}$

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Example Question #782 : Problem Solving Questions

Tetra_1

Refer to the above diagram. The perimeter of $\bigtriangleup DEG$ is 30. What is the surface area of the cube shown?

Possible Answers:

300

$600 \sqrt{2}$

Insufficient information is given to answer the question.

600

$300 \sqrt{2}$

Correct answer:

300

Explanation:

Since each side of $\bigtriangleup DEG$ is a diagonal of one of three congruent squares, each side has the same length; since the total perimeter is 30, each side measures one third of this, or 10.

Each square side, therefore, has diagonal 10, and, by the 45-45-90 Theorem, each side of each square - and each edge of the cube - measures $s= \frac{10}{\sqrt{2}}$ . We use the surface area formula:

$A= 6s^{2}$

$A= 6 \left ( \frac{10}{\sqrt{2}} \right )^{2} = 6 \cdot \frac{100}{2} = 300$

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Example Question #783 : Problem Solving Questions

A right prism has as its bases two triangles, each of which has a hypotenuse of length 25 and a leg of length 7. The height of the prism is one fourth the perimeter of a base. Give the surface area of the prism.

Possible Answers:

424

868

434

784

952

Correct answer:

952

Explanation:

The second leg of a right triangle with hypotenuse of length 25 and one leg of length 7 has length

$\sqrt{25^{2}-7^{2}}= \sqrt{625-49}= \sqrt{576}= 24$ .

The area of this right triangle is half the product of the lengths of the legs, which is

$B = \frac{1}{2} \cdot 7 \cdot 24 = 84$ .

The perimeter of each base is

P = 7+ 24 + 25 = 56 ,

and the height is one fourth this, or

$h = \frac{1}{4} \cdot P = \frac{1}{4} \cdot 56 =14$

The lateral area of the prism is the product of its height and the perimeter of a base; this is

$L = 14 \cdot 56 = 784$ .

The surface area is the sum of the lateral area and the two bases:

S= L + B + B = 784+84+84 =952 .

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Example Question #15 : Prisms

The length of a cube is increased by 20%, and the width is decreased by 20%. Which of the following must happen to the height so that the resulting rectangular prism will have the same surface area as the original cube?

Possible Answers:

The height must be increased by 4%.

The height must remain the same.

The height must be decreased by 4%.

The height must be decreased by 2%.

The height must be increased by 2%.

Correct answer:

The height must be increased by 2%.

Explanation:

To look at this more easily, assume the cube has sides of length 100; this argument generalizes to any size. The surface area of the cube is

$S= 6s^{2}= 6 \cdot 100^{2}= 6 \cdot 10,000 = 60,000$ .

After the changes, the resulting rectangular prism will have length

100 (1+0.2) = 120

and width

100 (1-0.2) = 80

The surface area of a rectangular prism is

S = 2LH + 2LW + 2WH . We can call L = 80 , W = 120 , and S= 60,000 and solve for :

2(80)H + 2(80)(120)+ 2(120)H= 60,000

160H + 19,200+240H= 60,000

400H + 19,200 = 60,000

400H = 40,800

H = 102

This means that the height of the prism must be 102% of the height of the cube - equivalently, the height must be increased by 2%.

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GMAT Math : Calculating the surface area of a prism

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Prisms

Example Question #61 : Rectangular Solids & Cylinders

Example Question #782 : Problem Solving Questions

Example Question #783 : Problem Solving Questions

Example Question #15 : Prisms

All GMAT Math Resources

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