How to find the volume of a pyramid - Math

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Question

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

Answer

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = $x^{2}$; h = 2x$

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

(b) $B = (2 $x)^{2}$ = $4x^{2}$; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $4x^{2}$ \cdot x = \frac {4}{3} $x^{3}$$

Regardless of , (b) is the greater quantity.

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Question

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Answer

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

$V = $\frac{1}{3}$ Bh = $\frac{1}{3}$ $s^{2}$h$ .

(a) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $1^{2}$ \cdot 4 = $\frac{4}{3}$$

(b) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $2^{2}$ \cdot 1 = $\frac{4}{3}$$

The two pyramids have equal volume.

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Question

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Answer

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

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Question

A pyramid with a square base has height equal to the perimeter of its base. Its volume is . In terms of , what is the length of each side of its base?

Answer

The volume of a pyramid is given by the formula

$V = $\frac{1}{3}$ A h$

where is the area of its base and is its height.

Let be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

and the area of the base is

$A = $s^{2}$$

So the volume formula becomes

$$\frac{1}{3}$ \cdot $s^{2}$ \cdot 4s = V$

Solve for :

$$\frac{4}{3}$ \cdot $s^{3}$ = V$

$$\frac{3}{4}$ \cdot $\frac{4}{3}$ \cdot $s^{3}$ =\frac{3}{4}$ \cdot V$

$s =\sqrt[3]{$\frac{3V}{4}$}$

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Question

An architect wants to make a square pyramid and fill it with 12,000 cubic feet of sand. If the base of the pyramid is 30 feet on each side, how tall does he need to make it?

Answer

Volume of Pyramid = 1/3 * Area of Base * Height

12,000 ft3 = 1/3 * 30ft * 30ft * H

12,000 = 300 * H

H = 12,000 / 300 = 40

H = 40 ft

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Question

The volume of a 6-foot-tall square pyramid is 8 cubic feet. How long are the sides of the base?

Answer

Volume of a pyramid is

$$\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (height)$

Thus:

$8=\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (6)$

$8=2\cdot (Area\ of\ the\ base)$

Area of the base is $4\ $ft^{2}$$ .

Therefore, each side is $2\ ft$ .

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Question

What is the sum of the number of vertices, edges, and faces of a square pyramid?

Answer

A square pyramid has one square base and four triangular sides.

Vertices (where two or more edges come together): 5. There are four vertices on the base (one at each corner of the square) and a fifth at the top of the pyramid.

Edges (where two faces come together): 8. There are four edges on the base (one along each side) and four more along the sides of the triangular faces extending from the corners of the base to the top vertex.

Faces (planar surfaces): 5. The base is one face, and there are four triangular faces that form the top of the pyramid.

Total

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Question

What is the volume of a pyramid with a height of and a square base with a side length of ?

Answer

To find the volume of a pyramid we must use the equation

$Volume: of: pyramid=\frac{1}{3}$(area\ of\ base)(height)$

We must first solve for the area of the square using

$A=(side\ $length^{2}$)$

We plug in and square it to get

We then plug our answer into the equation for the pyramid with the height to get

$V=\frac{1}{3}$(49)(15)$

We multiply the result to get our final answer for the volume of the pyramid

.

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Question

Find the volume of the following pyramid. Round the answer to the nearest integer.

Answer

The formula for the volume of a pyramid is:

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

where is the width of the base, is the length of the base, and is the height of the pyramid.

In order to determine the height of the pyramid, you will need to use the Pythagoream Theorem to find the slant height:

Now we can use the slant height to find the pyramid height, once again using the Pythagoream Theorem:

$A = $\sqrt{119}$m$

Plugging in our values, we get:

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

$V = $\frac{(10m)(10m)(\sqrt{119}$m)}{3}$

$V = $$\frac{100\sqrt{119}$m^3$}{3} \approx $364m^3$$

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Question

Find the volume of the following pyramid.

Answer

The formula for the volume of a pyramid is:

$V = $\frac{1}{3}$ (base)(height)$

$V = $\frac{1}{3}$ (l)(w)(h)$

Where is the length of the base, is the width of the base, and is the height of the pyramid

Use the Pythagorean Theorem to find the length of the slant height:

Now, use the Pythagorean Theorem again to find the length of the height:

$A = 2$\sqrt{7}$m$

Plugging in our values, we get:

$V = $\frac{1}{3}$ (12m)(12m)(2$\sqrt{7}$m)$

$V = $96$\sqrt{7}$m^3$$

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Question

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

Answer

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = $x^{2}$; h = 2x$

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

(b) $B = (2 $x)^{2}$ = $4x^{2}$; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot $4x^{2}$ \cdot x = \frac {4}{3} $x^{3}$$

Regardless of , (b) is the greater quantity.

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Question

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Answer

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

$V = $\frac{1}{3}$ Bh = $\frac{1}{3}$ $s^{2}$h$ .

(a) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $1^{2}$ \cdot 4 = $\frac{4}{3}$$

(b) Substitute : $V = $\frac{1}{3}$ $s^{2}$h = $\frac{1}{3}$ \cdot $2^{2}$ \cdot 1 = $\frac{4}{3}$$

The two pyramids have equal volume.

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Question

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Answer

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

Compare your answer with the correct one above

Question

A pyramid with a square base has height equal to the perimeter of its base. Its volume is . In terms of , what is the length of each side of its base?

Answer

The volume of a pyramid is given by the formula

$V = $\frac{1}{3}$ A h$

where is the area of its base and is its height.

Let be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

and the area of the base is

$A = $s^{2}$$

So the volume formula becomes

$$\frac{1}{3}$ \cdot $s^{2}$ \cdot 4s = V$

Solve for :

$$\frac{4}{3}$ \cdot $s^{3}$ = V$

$$\frac{3}{4}$ \cdot $\frac{4}{3}$ \cdot $s^{3}$ =\frac{3}{4}$ \cdot V$

$s =\sqrt[3]{$\frac{3V}{4}$}$

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Question

An architect wants to make a square pyramid and fill it with 12,000 cubic feet of sand. If the base of the pyramid is 30 feet on each side, how tall does he need to make it?

Answer

Volume of Pyramid = 1/3 * Area of Base * Height

12,000 ft3 = 1/3 * 30ft * 30ft * H

12,000 = 300 * H

H = 12,000 / 300 = 40

H = 40 ft

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Question

The volume of a 6-foot-tall square pyramid is 8 cubic feet. How long are the sides of the base?

Answer

Volume of a pyramid is

$$\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (height)$

Thus:

$8=\frac{1}{3}$\cdot (Area\ of\ the\ base)\cdot (6)$

$8=2\cdot (Area\ of\ the\ base)$

Area of the base is $4\ $ft^{2}$$ .

Therefore, each side is $2\ ft$ .

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Question

What is the sum of the number of vertices, edges, and faces of a square pyramid?

Answer

A square pyramid has one square base and four triangular sides.

Vertices (where two or more edges come together): 5. There are four vertices on the base (one at each corner of the square) and a fifth at the top of the pyramid.

Edges (where two faces come together): 8. There are four edges on the base (one along each side) and four more along the sides of the triangular faces extending from the corners of the base to the top vertex.

Faces (planar surfaces): 5. The base is one face, and there are four triangular faces that form the top of the pyramid.

Total

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Question

What is the volume of a pyramid with a height of and a square base with a side length of ?

Answer

To find the volume of a pyramid we must use the equation

$Volume: of: pyramid=\frac{1}{3}$(area\ of\ base)(height)$

We must first solve for the area of the square using

$A=(side\ $length^{2}$)$

We plug in and square it to get

We then plug our answer into the equation for the pyramid with the height to get

$V=\frac{1}{3}$(49)(15)$

We multiply the result to get our final answer for the volume of the pyramid

.

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Question

Find the volume of the following pyramid. Round the answer to the nearest integer.

Answer

The formula for the volume of a pyramid is:

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

where is the width of the base, is the length of the base, and is the height of the pyramid.

In order to determine the height of the pyramid, you will need to use the Pythagoream Theorem to find the slant height:

Now we can use the slant height to find the pyramid height, once again using the Pythagoream Theorem:

$A = $\sqrt{119}$m$

Plugging in our values, we get:

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

$V = $\frac{(10m)(10m)(\sqrt{119}$m)}{3}$

$V = $$\frac{100\sqrt{119}$m^3$}{3} \approx $364m^3$$

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Question

Find the volume of the following pyramid.

Answer

The formula for the volume of a pyramid is:

$V = $\frac{1}{3}$ (base)(height)$

$V = $\frac{1}{3}$ (l)(w)(h)$

Where is the length of the base, is the width of the base, and is the height of the pyramid

Use the Pythagorean Theorem to find the length of the slant height:

Now, use the Pythagorean Theorem again to find the length of the height:

$A = 2$\sqrt{7}$m$

Plugging in our values, we get:

$V = $\frac{1}{3}$ (12m)(12m)(2$\sqrt{7}$m)$

$V = $96$\sqrt{7}$m^3$$

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