Finding the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See the figure below.

An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r.

The height of a cone is the distance between its base and the vertex.

Earlier in this section, we saw that the volume of a cylinder is \(V=\text{π}{r}^{2}h.\) We can think of a cone as part of a cylinder. The figure below shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

An image of a cone is shown. There is a cylinder drawn around it.

The volume of a cone is less than the volume of a cylinder with the same base and height.

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

The formula V equals one-third times capital B times h is shown.

Since the base of a cone is a circle, we can substitute the formula of area of a circle, \(\text{π}{r}^{2}\) , for \(B\) to get the formula for volume of a cone.

The formula V equals one-third times pi times r squared times h is shown.

In this book, we will only find the volume of a cone, and not its surface area.

Definition: Volume of a Cone

For a cone with radius \(r\) and height \(h\).

An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h.

Example

Find the volume of a cone with height \(6\) inches and radius of its base \(2\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for \(\pi \))	\(V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{1.9em}{0ex}}{r}^{2}\phantom{\rule{1.7em}{0ex}}h\) \(V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2\right)}^{2}\phantom{\rule{1em}{0ex}}\left(6\right)\)
Step 5. Solve.	\(V\approx 25.12\)
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 25.12 cubic inches.

Example

Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is \(8\) inches tall and \(5\) inches in diameter? Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
Step 2. Identify what you are looking for.	the volume of the cone
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for \(\pi \), and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)	\(V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{2.3em}{0ex}}{r}^{2}\phantom{\rule{2em}{0ex}}h\) \(V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2.5\right)}^{2}\phantom{\rule{1em}{0ex}}\left(8\right)\)
Step 5. Solve.	\(V\approx 52.33\)
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume of the wrap is approximately 52.33 cubic inches.

Optional Video: Volume of a Cone

Finding the Volume of Cones