How to find the volume of a pyramid - ISEE Upper Level Quantitative Reasoning

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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 :

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

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

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

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

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Question

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Answer

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet = $4 \times 12 = 48$ inches

Sidelength of the base: 3 feet = $3 \times 12 = 36$ inches

The volume of a pyramid is

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

Since the base is a square, we can replace $B = s^{2}$ :

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

Substitute

$V = \frac{1}{3} \cdot 36 ^{2}\cdot 48$

The pyramid has volume 20,736 cubic inches.

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Question

A foot tall pyramid has a square base measuring feet on each side. What is the volume of the pyramid?

Answer

In order to find the area of a triangle, we use the formula $\frac{1}{3}Bh$ . In this case, since the base is a square, we can replace with $s^{2}$ , so our formula for volume is $\frac{1}{3}s^{2}h$ .

Since the length of each side of the base is feet, we can substitute it in for .

$\frac{1}{3}\cdot (40)^{2}\cdot h$

We also know that the height is feet, so we can substitute that in for .

$\frac{1}{3}\cdot (40)^{2}\cdot 300$

This gives us an answer of $160,000 ft^{3}$ .

It is important to remember that volume is expressed in units cubed.

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Question

The height of a right pyramid is inches. Its base is a square with sidelength inches. Give its volume in cubic feet.

Answer

Convert each of the measurements from inches to feet by dividing by .

Height: $42 \div 12 = 3 \frac{1}{2}$ feet

Sidelength: $24 \div 12 = 2$ feet

The base of the pyramid has area

$B = s^{2} = 2^{2} = 4$ square feet.

Substitute $B = 4, h = 3 \frac{1}{2}$ into the volume formula:

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

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

$V = \frac {1}{3} \cdot \frac {4}{1} \cdot \frac{7}{2}$

$V = \frac {1}{3} \cdot \frac {2}{1} \cdot \frac{7}{1} = \frac {14}{3} = 4 \frac {2}{3}$ cubic feet

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Question

The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.

Answer

Convert each of the measurements from feet to inches by multiplying by .

Height: $2 \times 12 = 24$ inches

Sidelength of base: $3 \times 12 = 36$ inches

The base of the pyramid has area

$B = s^{2} = 36^{2} = 1,296$ square inches.

Substitute into the volume formula:

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

$V = \frac {1}{3} \cdot 1,296 \cdot 24 =10,368$ cubic inches

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Question

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

Answer

The perimeter of the square base, feet, is equivalent to $3 \times 12 = 36$ inches; divide by to get the sidelength of the base - and the height: $36 \div 4 = 9$ inches.

The area of the base is therefore $B = s^{2} = 9^{2} = 81$ square inches.

In the formula for the volume of a pyramid, substitute :

$V = \frac{1}{3} Bh = \frac{1}{3}\cdot 81 \cdot 9 = 243$ cubic inches.

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Question

What is the volume of a pyramid with the following measurements?

$length=7; width=6;height=9;slant\ length=11$

Answer

The volume of a pyramid can be determined using the following equation:

$V=\frac{1}{3}lwh=\frac{1}{3}(7)(6)(9)=126$

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Question

A right regular pyramid with volume has its vertices at the points

where .

Evaluate .

Answer

The pyramid has a square base that is units by units, and its height is units, as can be seen from this diagram,

The square base has area $B = n^{2}$ ; the pyramid has volume $V = \frac{1}{3} \cdot n^{2} \cdot n = \frac{1}{3} n^{3}$

Since the volume is 1,000, we can set this equal to 1,000 and solve for :

$\frac{1}{3} n^{3} = 1,000$

$n^{3} = 3,000$

$n = \sqrt[3]{3,000} = \sqrt[3]{1,000} \cdot \sqrt[3]{3 } = 10 \sqrt[3]{3 }$

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Question

Find the volume of a pyramid with the following measurements:

length = 4in

width = 3in

height = 5in

Answer

To find the volume of a pyramid, we will use the following formula:

$V = \frac{l \cdot w \cdot h}{3}$

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the base of the pyramid has a length of 4in. We also know the base of the pyramid has a width of 3in. We also know the pyramid has a height of 5in.

Knowing this, we can substitute into the formula. We get

$V = \frac{4\text{in} \cdot 3\text{in} \cdot 5\text{in}}{3}$

$V = \frac{60\text{in}^3}{3}$

$V = 20\text{in}^3$

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Question

Find the volume of a pyramid with the following measurements:

length = 4cm

width = 9cm

height = 8cm

Answer

To find the volume of a pyramid, we will use the following formula:

$V = \frac{l \cdot w \cdot h}{3}$

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the following measurements:

length = 4cm

width = 9cm

height = 8cm

Knowing this, we can substitute into the formula. We get

$V = \frac{4\text{cm} \cdot 9\text{cm} \cdot 8\text{cm}}{3}$

$V = \frac{288\text{cm}^3}{3}$

$V = 96\text{cm}^3$

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Question

Find the volume of a pyramid with the following measurements:

length: 7in

width: 6in

height: 8in

Answer

To find the volume of a pyramid, we will use the following formula:

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

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in

width: 6in

height: 8in

So, we get

$V = \frac{ 7\text{in} \cdot 6\text{in} \cdot 8\text{in}}{3}$

$V = \frac{42\text{in}^2 \cdot 8\text{in}}{3}$

$V = \frac{336\text{in}^3}{3}$

$V = 112\text{in}^3$

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Question

Find the volume of a pyramid with the following measurements: length= in, width= in, height= in

Answer

To find the volume of a pyramid, we will use the following formula:

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

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in

width: 6in

height: 8in

So, we get

$V = \frac{ 7\text{in} \cdot 6\text{in} \cdot 8\text{in}}{3}$

$V = \frac{42\text{in} \cdot 8\text{in}}{3}$

$V = \frac{336\text{in}}{3}$

$V = 112\text{in}^3$

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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 :

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

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

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

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

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Question

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Answer

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet = $4 \times 12 = 48$ inches

Sidelength of the base: 3 feet = $3 \times 12 = 36$ inches

The volume of a pyramid is

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

Since the base is a square, we can replace $B = s^{2}$ :

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

Substitute

$V = \frac{1}{3} \cdot 36 ^{2}\cdot 48$

The pyramid has volume 20,736 cubic inches.

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