Finding the Missing Radius or Diameter of a Cone

📝 Practice Worksheets

Find the Missing Radius Cylinder Volume in Terms of Pi Volume of Cones Volume of Cylinders

🛠️ Related Tools

Volume Builder

🔑 Key Concepts

The volume of a cone is calculated using \(\displaystyle V=\frac{1}{3}\pi r^2h\).
When volume and height are known, substitute those values and solve the equation for \(r^2\).
Take the positive square root to find the radius because a geometric length must be positive.
If the problem asks for the diameter, multiply the radius by \(2\).
When the volume is written in terms of \(\pi\), the factor of \(\pi\) can be divided from both sides of the equation.

✏️ Worked Examples

Example 1: Find the Radius

A right cone has a volume of \(192\pi\) cubic centimeters and a height of \(16\) centimeters. Find the radius.

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

\(\displaystyle 192\pi=\frac{1}{3}\pi r^2(16)\)

Multiply both sides by \(3\):

\(\displaystyle {\color{red}{3}}(192\pi) = {\color{red}{3}}\left(\frac{16\pi r^2}{3}\right) \)

\(576\pi=16\pi r^2\)

Divide both sides by \(16\pi\):

\(\displaystyle \frac{576\pi}{\color{red}{16\pi}} = \frac{16\pi r^2}{\color{red}{16\pi}} \)

\(36=r^2\)

Take the positive square root of both sides:

\(\displaystyle {\color{red}{\sqrt{\phantom{36}}}}\;36 = {\color{red}{\sqrt{\phantom{r^2}}}}\;r^2 \)

\(6=r\)

Radius:
\(\boxed{r=6\text{ cm}}\)

Example 2: Find the Diameter

A right cone has a volume of \(343\pi\) cubic centimeters and a height of \(21\) centimeters. Find the diameter of its circular base.

\(\displaystyle 343\pi=\frac{1}{3}\pi r^2(21)\)

Simplify \(\displaystyle \frac{21}{3}\):

\(343\pi=7\pi r^2\)

Divide both sides by \(7\pi\):

\(\displaystyle \frac{343\pi}{\color{red}{7\pi}} = \frac{7\pi r^2}{\color{red}{7\pi}} \)

\(49=r^2\)

Take the positive square root:

\(\displaystyle {\color{red}{\sqrt{\phantom{49}}}}\;49 = {\color{red}{\sqrt{\phantom{r^2}}}}\;r^2 \)

\(7=r\)

The problem asks for the diameter:

\(d=2r\)

\(d=2(7)\)

\(d=14\)

Diameter:
\(\boxed{d=14\text{ cm}}\)

Example 3: Another Missing Diameter

A cone has a volume of \(300\pi\) cubic inches and a height of \(25\) inches. Find the diameter of the cone.

\(\displaystyle 300\pi=\frac{1}{3}\pi r^2(25)\)

Multiply both sides by \(3\):

\(\displaystyle {\color{red}{3}}(300\pi) = {\color{red}{3}}\left(\frac{25\pi r^2}{3}\right) \)

\(900\pi=25\pi r^2\)

Divide both sides by \(25\pi\):

\(\displaystyle \frac{900\pi}{\color{red}{25\pi}} = \frac{25\pi r^2}{\color{red}{25\pi}} \)

\(36=r^2\)

Take the positive square root:

\(\displaystyle {\color{red}{\sqrt{\phantom{36}}}}\;36 = {\color{red}{\sqrt{\phantom{r^2}}}}\;r^2 \)

\(6=r\)

Double the radius to find the diameter:

\(d=2r\)

\(d=2(6)=12\)

Diameter:
\(\boxed{d=12\text{ in.}}\)

General Process for Finding a Missing Radius or Diameter

Write the cone volume formula: \(\displaystyle V=\frac{1}{3}\pi r^2h\).
Substitute the known volume and height.
Use inverse operations to isolate \(r^2\).
Take the positive square root to find \(r\).
If the problem asks for diameter, use \(d=2r\).

🧠 Math Vocabulary

Cone: A three-dimensional solid with one circular base that narrows to a single vertex.
Radius: The distance from the center of the circular base to its outer edge.
Diameter: A segment passing through the center of a circle with endpoints on the circle; it is twice the radius.
Height: The perpendicular distance from the circular base of a cone to its vertex.
Volume: The amount of three-dimensional space contained inside a solid.
Cubic units: Units used to measure volume, such as cubic centimeters or \(\text{cm}^3\).
Square root: A value that, when multiplied by itself, produces a given number.
Inverse operation: An operation used to undo another operation.
Isolate the variable: Use algebraic operations to place a variable alone on one side of an equation.
Positive root: The positive square-root solution used for a geometric measurement.

💡 Main Idea

When the volume and height of a cone are known, the cone volume formula can be treated as an equation and solved for the missing radius. After isolating \(r^2\), take the positive square root to find the radius. If the requested measurement is the diameter, multiply the radius by \(2\).

📚 What You Should Already Know

Students should know the cone volume formula, how to substitute known values, simplify expressions containing \(\pi\), solve equations using multiplication and division, evaluate square roots, and use the relationship \(d=2r\).

🚀 What Comes Next

Students can extend this reasoning by solving for a missing cone height, finding unknown dimensions when volume is given as a decimal, comparing different cones with equal volumes, and solving missing-dimension problems involving cylinders and spheres.

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