Pre-Algebra - Volume of a Cone

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

$50\pi$

$125\pi$

$150\pi$

$450\pi$

$10\pi$

Explanation

Write the formula to find the volume of a cone.

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

Substitute the dimensions and solve.

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

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

$4\pi$

$12\pi$

$3\pi$

$6\pi$

$2\pi$

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.

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

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

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

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

$303\pi$

$300\pi$

$30\pi$

Explanation

Write the volume formula for a cone.

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

Substitute the dimensions.

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

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

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

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

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

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

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

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.

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

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

$39\pi$

$117\pi$

$13\pi$

$16\pi$

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

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.

$V_{cone} = 39\pi$

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

$\frac{2}{3}$

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

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

$2\pi$

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.

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

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

$4\pi$

$6\pi$

$18\pi$

$24\pi$

$\pi$

Explanation

Write the formula to find the volume of the cone.

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

Substitute the radius and height.

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

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

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

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

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

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

$6\pi\textup{ feet}$

Explanation

Write the formula to find the volume of a cone.

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

Volume of a Cone

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