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Pre-Algebra : Volume of a Cone

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Example Question #1 : Volume Of A Cone

The standard waffle cone at Cream Canyon Ice Cream Parlor has a diameter of 4 inches. If the height of the cone is 1.5 times the diameter, what is the volume of the cone?

Possible Answers:

$8\pi in.^{3}$

$48\pi in.^{3}$

$12\pi in.^{3}$

$4\pi in.^{3}$

$24\pi in.^{3}$

Correct answer:

$8\pi in.^{3}$

Explanation:

We must first recall the formula for the volume of a cone.

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

where is the radius and is the height. The problem is that we are not provided with either the height or the radius. However, we are told that the height is 1.5 times the diameter. Since the diameter is 4 inches, we can calculate $1.5\cdot4=6$ . Thus the height of the cone is 6 inches. We must also recall that the radius of a circle (such as the top of an ice cream cone) is simply half the diameter. Therefore, if the diameter is 4 inches, the radius is 2 inches. We now have all of the essential ingredients for volume.

$V=\frac{1}{3}\pi (2)^{2}(6)= \frac{1}{3}\pi (24)=8\pi$

Since our measurements were all in inches, our volume will be in cubic inches. Therefore, the volume of our ice cream cone is $8\pi in.^3$ . Now all we need is a scoop or two of our favorite flavor.

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Example Question #2 : Volume Of A Cone

Cone_1

Which of the following is the volume of the above cone?

Possible Answers:

$1,250 \pi$ cubic centimeters

$\frac{2,500 \pi }{3}$ cubic centimeters

$2,500 \pi$ cubic centimeters

$\frac{6,250\pi }{3}$ cubic centimeters

$6,250 \pi$ cubic centimeters

Correct answer:

$\frac{2,500 \pi }{3}$ cubic centimeters

Explanation:

The volume of a cone whose height is and whose base has radius is defined by the formula

$V = \frac{1}{3} \pi r^{2}h$ .

Set r = 10, h= 25 :

$V = \frac{1}{3} \pi \cdot 10 ^{2} \cdot 25$

$V = \frac{1}{3} \pi \cdot 100 \cdot 25$

$V = \frac{1}{3} \pi \cdot 2,500$

$V = \frac{2,500 \pi }{3}$ cubic centimeters.

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Example Question #1 : Volume Of A Cone

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ \sqrt[3]{ \pi^{2} }}{ 3 \pi}$

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 12 \sqrt[3]{ \pi^{2} }}{ \pi}$

$\frac{ 3 \sqrt[3]{ \pi^{2} }}{ \pi}$

Correct answer:

$\frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

Explanation:

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

$V =\frac{1}{3} \pi r^{2} \cdot 3r$

$V = \pi r^{3}$

Set this volume equal to one and solve for :

$\pi r^{3} = 1$

$\pi r^{3} \div \pi = 1 \div \pi$

$r^{3}=\frac{ 1 }{ \pi}$

$r=\sqrt[3]{\frac{ 1 }{ \pi}} ={\frac{ \sqrt[3] {1} }{ \sqrt[3]{ \pi}}} ={\frac{ 1}{ \sqrt[3]{ \pi}}}={\frac{1 \cdot \sqrt[3]{ \pi^{2} }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ \pi^{2}}} ={\frac{ \sqrt[3]{ \pi^{2} }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$\frac{ \sqrt[3]{ \pi^{2} }}{ \pi} \times 36 = \frac{ 36 \sqrt[3]{ \pi^{2} }}{ \pi}$

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Example Question #1 : Volume Of A Cone

What is the volume of a cone with a radius of two and a height of three?

Possible Answers:

$4\pi$

$18\pi$

$6\pi$

$\pi$

$24\pi$

Correct answer:

$4\pi$

Explanation:

Write the formula to find the volume of the cone.

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

Substitute the radius and height.

$V=\frac{1}{3}\pi (2)^2 (3) = 4\pi$

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Example Question #1 : Volume Of A Cone

What is the volume of a cone with a radius of 5 and a height of 6?

Possible Answers:

$125\pi$

$50\pi$

$10\pi$

$150\pi$

$450\pi$

Correct answer:

$50\pi$

Explanation:

Write the formula to find the volume of a cone.

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

Substitute the dimensions and solve.

$V=\frac{1}{3} \pi (5)^2 (6) = \pi (5)^2 (2) = 50 \pi$

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Example Question #6 : Volume Of A Cone

Find the volume of a cone with a base area of and a height of .

Possible Answers:

$\frac{2}{3}\pi$

$\frac{2}{3}$

$\frac{4}{9}\pi$

$2\pi$

Correct answer:

$\frac{2}{3}$

Explanation:

The base of a cone has a circular cross section. Given the base area, there is no need to determine the radius.

Write the formula for the volume of a cone.

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

The $\pi r^2$ term represents the base area of the circle.

Substitute all the given values into the volume formula.

$V=\frac{1}{3} (\pi r^2) h = \frac{1}{3} (1) (2) = \frac{2}{3}$

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Example Question #7 : Volume Of A Cone

What is the volume of a cone with diameter of 2 and a height of 10?

Possible Answers:

$\frac{40}{3}\pi$

$\frac{100}{3}\pi$

$\frac{10}{3}\pi$

$\frac{20}{3}\pi$

$\frac{200}{3}\pi$

Correct answer:

$\frac{10}{3}\pi$

Explanation:

Write the formula for the volume of a cone.

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

The radius is half the diameter.

Substitute the radius and the height.

$V=\frac{1}{3} \pi (1)^2(10) = \frac{10}{3}\pi$

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Example Question #8 : Volume Of A Cone

Find the volume of a cone with a radius of and a height of .

Possible Answers:

$\frac{100}{3}\pi$

$303\pi$

$\frac{101}{3}\pi$

$300\pi$

$30\pi$

Correct answer:

$\frac{100}{3}\pi$

Explanation:

Write the volume formula for a cone.

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

Substitute the dimensions.

$V=\frac{1}{3} \pi (10)^2 (1) =\frac{100}{3}\pi$

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Example Question #1 : Volume Of A Cone

Find the volume of the cone that has a radius of $\textup{3 feet}$ and a height of $\textup{2 feet}$ .

Possible Answers:

$6\pi\textup{ feet}^2$

$6\pi\textup{ feet}^3$

$18\pi\textup{ feet}^2$

$6\pi\textup{ feet}$

$18\pi\textup{ feet}^3$

Correct answer:

$6\pi\textup{ feet}^3$

Explanation:

Write the formula to find the volume of a cone.

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

$V=\frac{1}{3} \pi (3 \textup{ feet})^2 (2\textup{ feet}) =6\pi\textup{ feet}^3$

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Example Question #10 : Volume Of A Cone

Find the volume of a cone with a radius of and a height of .

Possible Answers:

$39\pi$

$117\pi$

$13\pi$

$16\pi$

$\frac{16}{3}\pi$

Correct answer:

$39\pi$

Explanation:

Write the formula to find the volume of the vone.

$V_{cone}= \frac{1}{3} \pi r^2h$

Substitute the dimensions into the formula.

$V_{cone}= \frac{1}{3} \pi (3)^2(13)$

Expand the terms.

$V_{cone}= \frac{1}{3} \pi (3\cdot3)(13) = \pi (3)(13)$

Multiply the integers for the volume.

$V_{cone} = 39\pi$

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Pre-Algebra : Volume of a Cone

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

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Example Question #1 : Volume Of A Cone

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Example Question #1 : Volume Of A Cone

Example Question #1 : Volume Of A Cone

Example Question #1 : Volume Of A Cone

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