Topics
7. 3D Geometry
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United States of AmericaVirginia
Grade 8

7.03 Volume of cones

Lesson
Practice

Volume of cones

We have previoulsy learned about cylinders and how to find their volume.

Cylinder

A solid figure formed by two congruent parallel faces called bases joined by a curved surface

A three dimensional solid with a pair of congruent circular bases, connected by a curved surface.

Recall we can find the volume of a cylinder using the formula:

V_\text{cylinder}=\pi r^{2}h

where r is the radius of the circular base and h is the height of the cylinder.

Now we will explore the relationship between cylinders and cones.

Exploration

In the following applet the cone and cylinder have the same radius and height.

Click the button to pour the water from the cone to the cylinder. Then press Refill to fill the cone again. and repeat until the cylinder is full.

Loading interactive...

  1. How many cones of water did it take to fill the cylinder?

  2. What fraction of the cylinder does one cone fill?

  3. Write a formula for the volume of a cone.

The volume of a cone is exactly one-third the volume of a cylinder formed from the same base with the same perpendicular height.

A cone with height 'h' and radius 'r'. The base is labeled with A=πr².

To find the volume of a cone we can use the formula:

V_{\text{cone}}=\dfrac{1}{3}\pi r^{2} h

where r is the radius of the circular base and h is the height of the cone.

Examples

Example 1

Find the volume of the cone shown. Round your answer to two decimal places.

A cone with radius of 2 meters and height of 6 meters. Ask you teacher for more information
Worked Solution
Create a strategy

Use the formula of the volume of cone.

Apply the idea

We have been given values for radius r=2 and height h=6.

\displaystyle V\displaystyle =\displaystyle \frac{1}{3}\pi r^2hUse the formula
\displaystyle =\displaystyle \frac{1}{3}\cdot\pi \cdot 2^{2}\cdot 6Substitute r = 2 and h = 6
\displaystyle =\displaystyle 25.13\, \operatorname{cm}^{3}Evaluate

Example 2

An ice cream cone has a volume of 6.28 \operatorname{ in}^{3} and a radius of 1 \operatorname{ in}. Find the height of the cone.

Worked Solution
Create a strategy

The volume of a cone is found by using the formula \dfrac{1}{3} \pi r^{2} h.

We can substitute the volume and radius that was given in the problem into the formula to solve for the missing variable h or the height of the cone. Use 3.14 to approximate \pi

Apply the idea

We are given that the volume is 6.28 \operatorname{in}^{3} and the radius is 1 \operatorname{in}. We can substitute those values for V \text{ and } r.

\displaystyle V\displaystyle =\displaystyle \dfrac{1}{3} \pi r^{2} hBegin with the formula for volume of a cylinder.
\displaystyle 6.28\displaystyle =\displaystyle \frac{1}{3} \cdot 3.14 \left(1\right)^{2} \cdot hSubstitute the given values and 3.14 for \pi
\displaystyle 6.28\displaystyle =\displaystyle \frac{1}{3} \cdot 3.14 \left(1\right) \cdot hEvaluate the exponent
\displaystyle 6.28\displaystyle =\displaystyle \dfrac{1}{3} \cdot 3.14 \cdot hEvaluate the multiplication
\displaystyle 18.84\displaystyle =\displaystyle 3.14 \cdot hMultiply both sides by 3
\displaystyle 6\displaystyle =\displaystyle hDivide both sides by 3.14

The height of the cone is 6 \operatorname{ in}.

Reflect and check

Our answer would have varied slightly if we had used \pi instead of 3.14.

\displaystyle V\displaystyle =\displaystyle \frac{1}{3} \pi r^{2} hBegin with the formula for volume of a cylinder.
\displaystyle 6.28\displaystyle =\displaystyle \dfrac{1}{3} \pi \left(1\right)^{2} hSubstitute the given values and 3.14 for \pi
\displaystyle 6.28\displaystyle =\displaystyle \dfrac{1}{3}\pi \left(1\right) hEvaluate the exponent
\displaystyle 6.28\displaystyle =\displaystyle \dfrac{1}{3} \pi hEvaluate the multiplication
\displaystyle 18.84\displaystyle =\displaystyle \pi hMultiply both sides by 3
\displaystyle 6\displaystyle =\displaystyle hDivide both sides by \pi

The height of the cone is 5.996 \operatorname{ in}.

Idea summary

The volume of a cone is exactly one-third the volume of a cylinder formed from the same base with the same perpendicular height.

To find the volume of a cone, we can use the formula:

\displaystyle V_\text{cone}=\dfrac{1}{3}\pi r^2h
\bm{V_{\text{cone}}}
Volume of the cone
\bm{r}
Radius of the cone
\bm{h}
Height of the cone

Outcomes

8.MG.2

The student will investigate and determine the surface area of square-based pyramids and the volume of cones and square-based pyramids.

8.MG.2b

Determine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas.

8.MG.2c

Examine and explain the relationship between the volume of cones and cylinders, and the volume of rectangular prisms and square based pyramids.

8.MG.2d

Solve problems in context involving volume of cones and square-based pyramids and the surface area of square-based pyramids.

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