Volume of a Pyramid - GED Math

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Question

A square pyramid whose base has sidelength has volume $2A^{3}$ . What is the ratio of the height of the pyramid to the sidelength of its base?

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.

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 $x^{2}$ . The height is . Therefore, setting $V = 100, B = x^{2}, h = x$ , we solve for in the equation

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

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

$x^{3} = 300$

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

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Question

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

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?

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?

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 .

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.

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.

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?

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.

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 $2A^{3}$ . What is the ratio of the height of the pyramid to the sidelength of its base?

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.

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 $x^{2}$ . The height is . Therefore, setting $V = 100, B = x^{2}, h = x$ , we solve for in the equation

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

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

$x^{3} = 300$

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

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Question

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

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?

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?

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:

Compare your answer with the correct one above

Question

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

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:

Compare your answer with the correct one above

Question

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

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:

Compare your answer with the correct one above

Question

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

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?

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:

Compare your answer with the correct one above

Question

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

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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