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Pre-Algebra : Volume of a Pyramid

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Example Question #1 : Volume Of A Pyramid

Principal O'Shaughnessy has a paperweight in the shape of a pyramid with a square base. If one side of the base has a length of 4cm and the height of the paperweight is 6cm, what is the volume of the paperweight?

Possible Answers:

32cm^3

48cm^3

192cm^3

576cm^3

96cm^3

Correct answer:

32cm^3

Explanation:

We begin by recalling the volume of a pyramid.

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

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

Since the base is a square, we can find the area by squaring the length of one of the sides.

A=4^2=16

Given the height is 6cm, we can now calculate the volume.

$V=\frac{1}{3}(16)(6)=32$

Since all of the measurements were in centimeters, our volume will be in cubic centimeters.

Therefore, the volume of Principal O'Shaughnessy's paperweight is 32cm^3 .

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Example Question #1 : Volume Of A Pyramid

The volume of a square pyramid is $162 m^{3}$ . If a side of the square base measures . What is the height of the pyramid?

Possible Answers:

7.5m

4.5m

Correct answer:

Explanation:

The formula for the volume of a pyramid is $\frac{\left ( B \times h\right )}{3}$ , where is the area of the base and is the height.

Using this formula,

= Area of the base, which is nothing but area of square with side . $9m \times 9m = 81 m^{2}$

Now, $\frac{81m^{2} \times h}{3} = 162 m^{3}$ when simplified, you get .

Hence, the height of the pyramid is .

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Example Question #1 : Volume

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

Possible Answers:

$5,184\textrm{ in}^{3}$

$20,736 \textrm{ in}^{3}$

$31,104\textrm{ in}^{3}$

$124,416\textrm{ in}^{3}$

$62,208 \textrm{ in}^{3}$

Correct answer:

$20,736 \textrm{ in}^{3}$

Explanation:

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 s=36, h = 48

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

V =20,736

The pyramid has volume 20,736 cubic inches.

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Example Question #2 : Volume

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

Possible Answers:

$31,104 \textrm{ in}^{3}$

$15,552 \textrm{ in}^{3}$

$41,472 \textrm{ in}^{3}$

$10,368 \textrm{ in}^{3}$

$62,208\textrm{ in}^{3}$

Correct answer:

$10,368 \textrm{ in}^{3}$

Explanation:

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 B = 1,296, h = 24 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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Example Question #2 : Volume

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.

Possible Answers:

$243 \textrm{ in}^{3}$

$81 \textrm{ in}^{3}$

$729 \textrm{ in}^{3}$

$576 \textrm{ in}^{3}$

$27 \textrm{ in}^{3}$

Correct answer:

$243 \textrm{ in}^{3}$

Explanation:

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 B = 81,h=9 :

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

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Example Question #3 : Volume

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

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

Possible Answers:

378

462

154

126

Correct answer:

126

Explanation:

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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Example Question #2 : Volume

The pyramid has a length, width, and height of 2,4,6 respectively. What is the volume of the pyramid?

Possible Answers:

Correct answer:

Explanation:

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (L\times W\times H)$

Substitute the dimensions and solve.

$V=\frac{1}{3} (2\times 4\times 6) = \frac{1}{3}(48)=16$

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Example Question #1 : Volume Of A Pyramid

If the base area of the pyramid is , and the height is , what is the volume of the pyramid?

Possible Answers:

Correct answer:

Explanation:

Write the volume formula for the pyramid.

$V=\frac{1}{3} (L\times W\times H)$

The base area is represented by $L\times W$ .

Substitute the knowns into the formula.

$V=\frac{1}{3} (2\times 3) = \frac{1}{3} (6) =2$

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Example Question #2 : Volume Of A Pyramid

Find the volume of a pyramid with a length of 4, width of 7, and a height of 3.

Possible Answers:

Correct answer:

Explanation:

Write the formula to find the area of a pyramid.

$V=\frac{1}{3} (L\times W \times H)$

Substitute the dimensions.

$V=\frac{1}{3} (4\times 7\times 3) = 28$

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Example Question #1 : Volume Of A Pyramid

Find the volume of a pyramid if the length, base, and height are $\textup{10, 20, and 40,}$ respectively.

Possible Answers:

400

$\frac{8000}{3}$

$\frac{400}{3}$

500

$\frac{800}{3}$

Correct answer:

$\frac{8000}{3}$

Explanation:

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (LWH)$

Substitute the dimensions and solve for the volume.

$V=\frac{1}{3} (10)(20)(40) = \frac{8000}{3}$

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Pre-Algebra : Volume of a Pyramid

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Example Question #1 : Volume Of A Pyramid

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