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Intermediate Geometry : Prisms

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Example Question #4 : How To Find The Diagonal Of A Prism

A right, rectangular prism has a length of meters, width that is meters longer than the length, and a height of meters. The volume of the prism is 100 cubic meters. Find the diagonal of the prism.

Possible Answers:

$\sqrt{129} m$

20 m

None of the other answers.

2 m

$3\sqrt{41} m$

Correct answer:

$\sqrt{129} m$

Explanation:

First, use the volume formula,

V = lwh to find the missing length and width.

$V = x\cdot(3+x)\cdot10\rightarrow 100 = (3x + x^{2})10$

$\rightarrow 100 = 30x + 10x^{2}$

$\rightarrow 10x^{2} + 30 x - 100 = 0 \rightarrow (5x - 10)(2x + 10) = 0$

$\rightarrow x = 2, -5$

Since cannot be a negative value is it represents a length of a prism, we know x=2 . So the length is 2 meters, and therefore the width is 5 meters.

Now you can plug in the length, width and height into the formula for finding the diagonal of the prism.

$d = \sqrt{2^{2} + 5^{2} + 10^{2}} = \sqrt{4 + 25 + 100} = \sqrt{129}$ .

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Example Question #5 : How To Find The Diagonal Of A Prism

A right, rectangular prism has a volume of cubic centimeters. Its width is cm and its length is three times its height. Find the length of the diagonal of the prism.

Possible Answers:

$2\sqrt{26} cm$

2 cm

$2\sqrt{6} cm$

$\sqrt{104} cm$

None of the other answers.

Correct answer:

$2\sqrt{26} cm$

Explanation:

First, use the volume to find the missing height and length.

Since the length is three times the height, use to represent the length and to represent the height.

So, $V = lwh \rightarrow 96 = 3x\cdot8\cdotx \rightarrow 96 = 24x^{2} \rightarrow 4 = x^2 \rightarrow 2 = x$ .

So the height of the prism is 2 centimeters, and the length is 6 centimeters.

Use these values to now solve for the diagonal distance.

$d = \sqrt{6^{2} + 8^{2} + 2^{2}} = \sqrt{36 + 64 + 4} = \sqrt{104} = 2\sqrt{26}$ .

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Example Question #6 : How To Find The Diagonal Of A Prism

The surface area of a right, rectangular prism is 280 square inches. The height is inches and the length is times the width. Find the diagonal distance of the prism.

Possible Answers:

2 in

$4\sqrt{51} in$

None of the other choices.

$\sqrt{204} in$

$2\sqrt{51} in$

Correct answer:

$2\sqrt{51} in$

Explanation:

Use the surface area, 280 square inches, and the formula for finding the surface area of a right, rectangular prsim SA = (2lw) + (2wh) + (2lh) to find the missing length and width measurements.

So,

280 = 2(5x)(x) + 2(5x)(10) + 2(x)(10)

$\rightarrow 280 = 10x^{2} + 100x + 20x \rightarrow 280 = 10x^{2}+ 120x$

$\rightarrow 10x^2 +120x - 280 = 0 \rightarrow 10(x^{2} + 12x - 28) = 0$

$\rightarrow 10(x-2)(x+14) = 0) \rightarrow x=2, -14$

Since represents the length of a solid figure, we must assume x=2 , rather than the negative value.

So, the width of the figure is 2 inches and the length is 10 inches.

Now, use the formula for finding the diagonal of a right, rectangular prism:

$d = \sqrt10^{2} + 2^{2} + 10^{2} = \sqrt{100 + 4 + 10} = \sqrt{204} = 2\sqrt{51}$ .

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Example Question #7 : How To Find The Diagonal Of A Prism

A right, rectangular prism has a surface area of square meters. Its width is twice its length, and its height is four times its length. Find the diagonal distance of the prism.

Possible Answers:

$3\sqrt{7} m$

1 m

$2\sqrt{12} m$

None of the other answers.

$\sqrt{21} m$

Correct answer:

$\sqrt{21} m$

Explanation:

Use the surface area of the prism to find the missing length, width and height.

$2(x)(2x) + 2(x)(4x) + 2(2x)(4x) = 28 \rightarrow 4x^{2} + 8x^{2} + 16x^{2} = 28$

$\rightarrow 28x^{2} = 28 \rightarrow x^{2} = 1 \rightarrow x = 1$

So, the prism's length is 1 meter, the width is 2 meters and the height is 4 meters.

Now you can find the diagonal distance using those values.

$d = \sqrt{1^{2} + 2^{2} + 4^{2}} = \sqrt{1 + 4 + 16} = \sqrt{21}$ .

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Example Question #8 : How To Find The Diagonal Of A Prism

Cube

The above cube has edges of length 1. True or false: The dashed line has length $\sqrt{2}$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Examine the diagram below.

Cube

$\bigtriangleup ABC$ is a right triangle with legs of length AB = AC = 1 , so it is an isosceles - or 45-45-90 - right triangle. By the 45-45-90 Triangle Theorem its hypotenuse measures

$AC = 1 \cdot \sqrt{2} = \sqrt{2}$

$\bigtriangleup DAC$ is a right triangle with legs of lengths AD = 1 and $AC = \sqrt{2}$ , so the length of its hypotenuse is

$DC = \sqrt{(AD)^{2}+ \left ( AC\right )^{2}}$

$DC = \sqrt{1^{2}+ \left ( \sqrt{2}\right )^{2}}$

$DC = \sqrt{1 +2}$

$DC = \sqrt{3}$ .

$\overline{DC}$ , the diagonal in question, has length $\sqrt{3}$ , not $\sqrt{2}$ .

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Intermediate Geometry : Prisms

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #4 : How To Find The Diagonal Of A Prism

Example Question #5 : How To Find The Diagonal Of A Prism

Example Question #6 : How To Find The Diagonal Of A Prism

Example Question #7 : How To Find The Diagonal Of A Prism

Example Question #8 : How To Find The Diagonal Of A Prism

All Intermediate Geometry Resources

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