Lesson 15: The Volume of a Cone

Let’s explore cones and their volumes.

15.1: Which Has a Larger Volume?

The cone and cylinder have the same height, and the radii of their bases are equal.

  1. Which figure has a larger volume?
  2. Do you think the volume of the smaller one is more or less than $\frac12$ the volume of the larger one? Explain your reasoning.
A right circular cylinder and a right circular cone. Both the cylinder and the cone have a height of 8.
  1. Here is a method for quickly sketching a cone: 
    A figure of three drawings. From left to right, the first drawing is of an oval. In the second drawing is of the oval and a point centered directly above the oval. The third drawing is of the oval and the point above the oval where two line segments are drawn from the point to the edges of the oval.
    • Draw an oval.
    • Draw a point centered above the oval.
    • Connect the edges of the oval to the point.
    • Which parts of your drawing would be hidden behind the object? Make these parts dashed lines.
    Sketch two different sized cones. The oval doesn’t have to be on the bottom! For each drawing, label the cone’s radius with $r$ and height with $h$.

15.2: From Cylinders to Cones

A right cirular cylinder and a right circular cone. Both the cylinder and the cone have a height labeled h and have a radius labeled r.

A cone and cylinder have the same height and their bases are congruent circles.

  1. If the volume of the cylinder is 90 cm3, what is the volume of the cone?
  2. If the volume of the cone is 120 cm3, what is the volume of the cylinder?
  3. If the volume of the cylinder is $V=\pi r^2h$, what is the volume of the cone? Either write an expression for the cone or explain the relationship in words.
     

15.3: Calculate That Cone

  1. Here is a cylinder and cone that have the same height and the same base area.

    What is the volume of each figure? Express your answers in terms of $\pi$.

    A right circular cylinder and a right circular cone. The cylinder has a diameter of 10 and a height of 4. In the cone, the height is indicated with dashed line and the radius is also indicated with a dashed line.
  2. Here is a cone.
    1. What is the area of the base? Express your answer in terms of $\pi$.
    2. What is the volume of the cone? Express your answer in terms of $\pi$.
    A right circular cone with a height of 8 and a radius of 6.
  3. A cone-shaped popcorn cup has a radius of 5 centimeters and a height of 9 centimeters. How many cubic centimeters of popcorn can the cup hold? Use 3.14 as an approximation for $\pi$, and give a numerical answer.

Summary

If a cone and a cylinder have the same base and the same height, then the volume of the cone is $\frac{1}{3}$ of the volume of the cylinder. For example, the cylinder and cone shown here both have a base with radius 3 feet and a height of 7 feet.

The cylinder has a volume of $63\pi$ cubic feet since $\pi \boldcdot 3^2 \boldcdot 7 = 63\pi$. The cone has a volume that is $\frac13$ of that, or $21\pi$ cubic feet.

An image of a right circular cone and a right circular cylinder. The cone has a height of 7 and radius of 3. The cylinder has a height of 7 and a radius of 3.

If the radius for both is $r$ and the height for both is $h$, then the volume of the cylinder is $\pi r^2h$. That means that the volume, $V$, of the cone is $$V=\frac{1}{3}\pi r^2h$$

Practice Problems ▶