Volume of a Pyramid - GED Math

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Question

A square pyramid whose base has sidelength has volume . What is the ratio of the height of the pyramid to the sidelength of its base?

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Answer

Since the correct answer is independent of the value of , for simplicity's sake, assume that .

The volume of a square pyramid with base of area and with height is

$V = $\frac{1}{3}$ Bh$ .

The base, being a square of sidelength 1, has area 1. In the volume formula, we set and $V = 2 $A^{3}$ = 2 \cdot $1^{3}$ = 2$ , and solve for :

$$\frac{1}{3}$ \cdot 1 \cdot h = 2$

$$\frac{1}{3}$ h = 2$

$3 \cdot $\frac{1}{3}$ h =3 \cdot 2$

This means that the height-to-sidelength ratio is equal to , which is the correct response.

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Question

The above square pyramid has volume 100. Evaluate to the nearest tenth.

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Answer

The volume of a square pyramid with base of area and with height is

$V = $\frac{1}{3}$ Bh$ .

The base, being a square of sidelength , has area . The height is . Therefore, setting $V = 100, B = $x^{2}$, h = x$ , we solve for in the equation

$$\frac{1}{3}$ $x^{2}$ \cdot x = 100$

$$\frac{1}{3}$ $x^{3}$ = 100$

$x = \sqrt[3]{300} \approx 6.7$

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Question

Suppose a triangular pyramid has base area of 10, and a height of 6. What is the volume?

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Answer

Write the formula for the volume of a pyramid.

$V= $\frac{B\times h}{3}$$

Substitute the known base area and the height into the formula.

$V= $\frac{10\times 6}{3}$ = $\frac{60}{3}$ = 20$

The answer is:

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Question

If the base of a pyramid is a square, with a length of 5, and the height of the pyramid is 9, what must be the volume?

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Answer

Write the formula for the volume of pyramid.

$V=\frac{Bh}{3}$$

The base area of a square is .

Substituting the side length:

Substitute the base area and the height.

$V=\frac{25\times 9}{3}$ = 25\times 3 = 75$

The answer is:

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Question

Suppose the base area of a pyramid is 24, and the height is 10. What must the volume be?

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Answer

Write formula for a pyramid.

$V=\frac{(LW)H}{3}$ = $\frac{BH}{3}$$

Substitute the base and height.

$V = $\frac{24\times 10}{3}$ = 8\times 10 = 80$

The answer is:

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Question

Find the volume of a pyramid with a square base area of and a height of .

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Answer

Write the formula of the volume of a pyramid.

$V = $\frac{LWH}{3}$$

The base area is represented by . This means the volume is:

$V = $\frac{LWH}{3}$ =\frac{BH}{3}$$

Substitute the base and height.

$V=\frac{6(10)}{3}$ = 20$

The answer is:

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Question

Find the volume of a square pyramid with a base area of 12 and a height of 4.

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Answer

Write the formula for the volume of a pyramid.

$V = $\frac{LWH}{3}$$

The base area constitutes of the given equation.

Substitute values into the formula.

$V = $\frac{12(4)}{3}$ = 4(4) = 16$

The answer is:

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Question

Find the volume of a square pyramid if the base area is 2, and the height is 5.

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Answer

Write the formula for the volume of a pyramid.

$V =\frac{LWH}{3}$$

The base area constitutes , and can be replaced with the numerical area.

$V =\frac{LWH}{3}$ =\frac{BH}{3}$ = $\frac{2\times 5}{3}$ =\frac{10}{3}$$

The answer is: $\frac{10}{3}$

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Question

If the length, width, and height of a pyramid is 2, 7, and 9, respectively, what must be the volume?

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Answer

Write the formula for the volume of a pyramid.

$V= $\frac{LWH}{3}$ = $\frac{2(7)(9)}{3}$ = 2(7)(3) = 42$

The answer is:

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Question

Find the volume of a pyramid with a length, width, and height of , respectively.

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Answer

Write the volume formula for the pyramid.

$V=\frac{LWH}{3}$$

Substitute the dimensions.

$V=\frac{2\times 5 \times 12}{3}$ = (2)(5)(4) = 40$

The answer is:

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Question

A square pyramid whose base has sidelength has volume . What is the ratio of the height of the pyramid to the sidelength of its base?

Tap to reveal answer

Answer

Since the correct answer is independent of the value of , for simplicity's sake, assume that .

The volume of a square pyramid with base of area and with height is

$V = $\frac{1}{3}$ Bh$ .

The base, being a square of sidelength 1, has area 1. In the volume formula, we set and $V = 2 $A^{3}$ = 2 \cdot $1^{3}$ = 2$ , and solve for :

$$\frac{1}{3}$ \cdot 1 \cdot h = 2$

$$\frac{1}{3}$ h = 2$

$3 \cdot $\frac{1}{3}$ h =3 \cdot 2$

This means that the height-to-sidelength ratio is equal to , which is the correct response.

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Question

The above square pyramid has volume 100. Evaluate to the nearest tenth.

Tap to reveal answer

Answer

The volume of a square pyramid with base of area and with height is

$V = $\frac{1}{3}$ Bh$ .

The base, being a square of sidelength , has area . The height is . Therefore, setting $V = 100, B = $x^{2}$, h = x$ , we solve for in the equation

$$\frac{1}{3}$ $x^{2}$ \cdot x = 100$

$$\frac{1}{3}$ $x^{3}$ = 100$

$x = \sqrt[3]{300} \approx 6.7$

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Question

Suppose a triangular pyramid has base area of 10, and a height of 6. What is the volume?

Tap to reveal answer

Answer

Write the formula for the volume of a pyramid.

$V= $\frac{B\times h}{3}$$

Substitute the known base area and the height into the formula.

$V= $\frac{10\times 6}{3}$ = $\frac{60}{3}$ = 20$

The answer is:

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Question

If the base of a pyramid is a square, with a length of 5, and the height of the pyramid is 9, what must be the volume?

Tap to reveal answer

Answer

Write the formula for the volume of pyramid.

$V=\frac{Bh}{3}$$

The base area of a square is .

Substituting the side length:

Substitute the base area and the height.

$V=\frac{25\times 9}{3}$ = 25\times 3 = 75$

The answer is:

← Didn't Know|Knew It →

Question

Suppose the base area of a pyramid is 24, and the height is 10. What must the volume be?

Tap to reveal answer

Answer

Write formula for a pyramid.

$V=\frac{(LW)H}{3}$ = $\frac{BH}{3}$$

Substitute the base and height.

$V = $\frac{24\times 10}{3}$ = 8\times 10 = 80$

The answer is:

← Didn't Know|Knew It →

Question

Find the volume of a pyramid with a square base area of and a height of .

Tap to reveal answer

Answer

Write the formula of the volume of a pyramid.

$V = $\frac{LWH}{3}$$

The base area is represented by . This means the volume is:

$V = $\frac{LWH}{3}$ =\frac{BH}{3}$$

Substitute the base and height.

$V=\frac{6(10)}{3}$ = 20$

The answer is:

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Question

Find the volume of a square pyramid with a base area of 12 and a height of 4.

Tap to reveal answer

Answer

Write the formula for the volume of a pyramid.

$V = $\frac{LWH}{3}$$

The base area constitutes of the given equation.

Substitute values into the formula.

$V = $\frac{12(4)}{3}$ = 4(4) = 16$

The answer is:

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Question

Find the volume of a square pyramid if the base area is 2, and the height is 5.

Tap to reveal answer

Answer

Write the formula for the volume of a pyramid.

$V =\frac{LWH}{3}$$

The base area constitutes , and can be replaced with the numerical area.

$V =\frac{LWH}{3}$ =\frac{BH}{3}$ = $\frac{2\times 5}{3}$ =\frac{10}{3}$$

The answer is: $\frac{10}{3}$

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Question

If the length, width, and height of a pyramid is 2, 7, and 9, respectively, what must be the volume?

Tap to reveal answer

Answer

Write the formula for the volume of a pyramid.

$V= $\frac{LWH}{3}$ = $\frac{2(7)(9)}{3}$ = 2(7)(3) = 42$

The answer is:

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Question

Find the volume of a pyramid with a length, width, and height of , respectively.

Tap to reveal answer

Answer

Write the volume formula for the pyramid.

$V=\frac{LWH}{3}$$

Substitute the dimensions.

$V=\frac{2\times 5 \times 12}{3}$ = (2)(5)(4) = 40$

The answer is:

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