Lesson 20Finding Cone Dimensions

Learning Goal

Let’s figure out the dimensions of cones.

Learning Targets

I can find missing information of about a cone if I know its volume and some other information.

Lesson Terms

cone
cylinder
sphere

Warm Up: Number Talk: Thirds

For each equation, decide what value, if any, would make it true.

$27 = \frac{1}{3} h$
$27 = \frac{1}{3} r^{2}$
$12 π = \frac{1}{3} π a$
$12 π = \frac{1}{3} π b^{2}$

Activity 1: An Unknown Radius

The volume $V$ of a cone with radius $r$ is given by the formula $V = \frac{1}{3} π r^{2} h$ .

A cone. The radius is labeled "r" and the height is labeled "3."

The volume of this cone with height 3 units and radius $r$ is $V = 64 π$ cubic units. This statement is true:

$64 π = \frac{1}{3} π r^{2} \cdot 3$ What does the radius of this cone have to be? Explain how you know.

Activity 2: Cones with Unknown Dimensions

A cone with three labeled measurements. A dashed line goes through the center of the circular base and touches two points on the base's edge and is labeled d. Another dashed line is drawn from the center of the circular base, touches the edge of the circular base, and is labeled r. A third dashed line is drawn from the center of the circular base to a point cenetred above the circular base and is labeled "h."

Each row of the table has some information about a particular cone. Complete the table with the missing dimensions.

diameter (units)	radius (units)	area of the base (square units)	height (units)	volume of cone (cubic units)
	$4$		$3$
	$\frac{1}{3}$		$6$
		$144 π$	$\frac{1}{4}$
$20$				$200 π$
			$12$	$64 π$
			$3$	$3.14$

Are you ready for more?

A frustum is the result of taking a cone and slicing off a smaller cone using a cut parallel to the base.

Find a formula for the volume of a frustum, including deciding which quantities you are going to include in your formula.

Activity 3: Popcorn Deals

A movie theater offers two containers:

An image of two containers of popcorn. The first container of popcorn is shaped like a cone. The distance from the edge of the opening to the point at the bottom is labeled 19 centimeters. The distance that passes through the center of the circlular base from one edge of the opening to the other edge of the opening is 12 centimeters. The price is labeled as 6 point 7 5 dollars. The second container of popcorn is shaped like a cylinder. The horizontal distance from the edge of the opening to the bottom of the container is 15 centimeters. The distance that passes through the center of the circular base from one edge of the opening to the other edge of the opening is 8 centimeters. The price is labeled as 6 point 2 5 dollars.

Which container is the better value? Use 3.14 as an approximation for $π$ .

Lesson Summary

As we saw with cylinders, the volume $V$ of a cone depends on the radius $r$ of the base and the height $h$ :

$V = \frac{1}{3} π r^{2} h$

If we know the radius and height, we can find the volume. If we know the volume and one of the dimensions (either radius or height), we can find the other dimension.

For example, imagine a cone with a volume of $64 π$ cm³, a height of 3 cm, and an unknown radius $r$ . From the volume formula, we know that

$64 π = \frac{1}{3} π r^{2} \cdot 3$

Looking at the structure of the equation, we can see that $r^{2} = 64$ , so the radius must be 8 cm.

Now imagine a different cone with a volume of $18 π$ cm³, a radius of 3 cm, and an unknown height $h$ . Using the formula for the volume of the cone, we know that

$18 π = \frac{1}{3} π 3^{2} h$

so the height must be 6 cm. Can you see why?

Lesson 20Finding Cone Dimensions

Learning Goal

Learning Targets

Lesson Terms

Warm Up: Number Talk: Thirds

Problem 1

Activity 1: An Unknown Radius

Problem 1

Activity 2: Cones with Unknown Dimensions

Problem 1

Are you ready for more?

Problem 1

Activity 3: Popcorn Deals

Problem 1

Lesson Summary