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Pre-Algebra : Geometry

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Example Questions

Example Question #8 : Volume Of A Cone

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

Possible Answers:

$300\pi$

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

$30\pi$

$303\pi$

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

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:

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

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

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

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

$6\pi\textup{ feet}$

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 #5 : Volume Of A Cone

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

Possible Answers:

$39\pi$

$13\pi$

$16\pi$

$117\pi$

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

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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Example Question #21 : Volume

Solve for the volume of a cone with a circular area of $2\pi$ and a height of .

Possible Answers:

$2\pi$

$6\pi$

$3\pi$

$4\pi$

$12\pi$

Correct answer:

$4\pi$

Explanation:

Write the formula for the volume of a cone.

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

The area of the circular base is represented as $\pi r^2$ in the equation.

Substitute $2\pi$ in replacement of this term. Substitute the height into the volume formula as well.

$V_\textup{cone} = \frac{1}{3} (\pi r^2)h = \frac{1}{3} (2\pi)(6)$

Multiply out each term and leave $\pi$ as is.

$V= \frac{1}{3} (2\pi)(6) = (2\pi)(2) = 4\pi$

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Example Question #22 : Volume

You have an ice cream cone. You want to fill the cone completely with ice cream. What is the volume of ice cream you can fill it with if the height of the cone is 12cm and the diameter is 8cm?

Possible Answers:

$96\pi \text{cm}^3$

$64\pi \text{cm}^3$

$128\pi \text{cm}^3$

$288\pi \text{cm}^3$

$48\pi \text{cm}^3$

Correct answer:

$64\pi \text{cm}^3$

Explanation:

The formula to find the volume of a cone is

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

where r is the radius and h is the height. We know the diameter of the ice cream cone is 8cm. We also know the radius is half the diameter, so the radius is 4cm. We know the height is 12cm. Using substitution, we get

$V = \pi \cdot (4\text{cm})^2 \cdot \frac{12\text{cm}}{3}$

$V = \pi \cdot 16\text{cm}^2 \cdot 4\text{cm}$

$V = \pi \cdot 64\text{cm}^3$

$V = 64\pi \ \text{cm}^3$

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Example Question #1 : Volume Of A Sphere

A certain children's basketball has a diameter of 6 inches. What is the volume of the basketball?

Possible Answers:

$18\pi in.^{3}$

$36\pi in.^{3}$

$9\pi in.^{3}$

$108\pi in.^{3}$

$288\pi in.^{3}$

Correct answer:

$36\pi in.^{3}$

Explanation:

Math involves a lot of formulas. Unfortunately that means we either have to do some memorization or compile a really long list of formulas for reference. Either way, a basketball is a sphere, which means we have to know the formula for the volume of a sphere to solve this problem.

$V=\frac{4}{3}\pi r^{3}$

stands for volume, and stands for radius. Therefore, all we need to do is plug in the radius of the basketball and do a little calculation to find our answer. The problem is that we do not have the radius. All we have is a diameter. But if we remember the relationship between diameters and radii (the plural of radius), we have nothing to fear. The diameter is just twice as long as the radius. That means that if our diameter is 6 inches, the radius is simply 3 inches. Now that we have our radius, we are ready to plug and play.

$V=\frac{4}{3}\pi (3)^{3}=\frac{4}{3}\pi(27)=36\pi$

Since our original unit was inches, and volume is always expressed in cubic units, our final answer is

$36\pi in.^3$

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Example Question #1 : Volume Of A Sphere

In terms of $\pi$ , give the volume, in cubic inches, of a spherical water tank with a diameter of 20 feet.

Possible Answers:

$2,304,000 \pi \textrm{ in}^{3}$

$307,200\pi\textrm{ in}^{3}$

$230,400 \pi\textrm{ in}^{3}$

$18,432,000\pi\textrm{ in}^{3}$

$57,600 \pi\textrm{ in}^{3}$

Correct answer:

$2,304,000 \pi \textrm{ in}^{3}$

Explanation:

20 feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set r = 120 , substitute in the volume formula, and solve for :

$V = \frac{4}{3} \pi r^{3}$

$V = \frac{4}{3} \pi \cdot 120^{3}$

$V = \frac{4}{3} \pi \cdot 1,728,000$

$V = 2,304,000 \pi$ cubic inches

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Example Question #2 : Volume Of A Sphere

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

Possible Answers:

$4,500,000\pi \textrm{ cm}^{3}$

$36,000,000\pi \textrm{ cm}^{3}$

$1,125,000\pi \textrm{ cm}^{3}$

$27,000,000\pi \textrm{ cm}^{3}$

$3,375,000\pi \textrm{ cm}^{3}$

Correct answer:

$4,500,000\pi \textrm{ cm}^{3}$

Explanation:

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute r = 150 in the volume formula:

$V= \frac{4}{3} \pi r^{3}$

$V= \frac{4}{3} \cdot \pi \cdot 150^{3}$

$V= \frac{4}{3} \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

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Example Question #4 : Volume Of A Sphere

A car dealership wants to fill a large spherical advertising ballon with helium. It can afford to buy 1,000 cubic yards of helium to fill this balloon. What is the greatest possible diameter of that balloon (nearest tenth of a yard)?

Possible Answers:

$7.7 \;\textrm{yd}$

$9.3 \;\textrm{yd}$

$15.4 \;\textrm{yd}$

$6.2 \;\textrm{yd}$

$12.4 \;\textrm{yd}$

Correct answer:

$12.4 \;\textrm{yd}$

Explanation:

The volume of a sphere, given its radius, is

$V = \frac{4\pi r^{3}}{3}$

Set V = 1,000 , solve for , and double that to get the diameter.

$1,000= \frac{4\pi r^{3}}{3}$

$r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$

$r \approx \sqrt[3]{238.7} \approx 6.2$

The diameter is twice this, or 12.4 yards.

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Example Question #1 : Volume Of A Sphere

The diameter of a sphere is . What is the volume of the sphere?

Possible Answers:

$\frac{33}{4}\pi$

$\frac{32}{3}\pi$

$\frac{18}{7}\pi$

$32\pi$

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

Correct answer:

$\frac{32}{3}\pi$

Explanation:

The formula for finding volume of a sphere is:

$V = \frac{4}{3}\pi r^{3}$

We are given the diameter, so to find radius we divide the diameter by :

$4\div2 = 2$

We plug this radius into the formula:

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

$V = \frac{4}{3}\pi \left ( 8\right )$

$V = \frac{32}{3}\pi$

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Pre-Algebra : Geometry

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #8 : Volume Of A Cone

Example Question #1 : Volume Of A Cone

Example Question #5 : Volume Of A Cone

Example Question #21 : Volume

Example Question #22 : Volume

Example Question #1 : Volume Of A Sphere

Example Question #1 : Volume Of A Sphere

Example Question #2 : Volume Of A Sphere

Example Question #4 : Volume Of A Sphere

Example Question #1 : Volume Of A Sphere

All Pre-Algebra Resources

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