# Lesson 20

Finding Cone Dimensions

Let’s figure out the dimensions of cones.

### 20.1: Number Talk: Thirds

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

\(27=\frac13h\)

\(27=\frac13r^2\)

\(12\pi=\frac13\pi a\)

\(12\pi=\frac13\pi b^2\)

### 20.2: An Unknown Radius

The volume \(V\) of a cone with radius \(r\) is given by the formula \(V=\frac13 \pi r^2h\).

The volume of this cone with height 3 units and radius \(r\) is \(V=64\pi\) cubic units. This statement is true:

\(\displaystyle 64\pi =\frac13 \pi r^2 \boldcdot 3\) What does the radius of this cone have to be? Explain how you know.

### 20.3: Cones with Unknown Dimensions

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 | |||

\(144\pi\) | \(\frac14\) | |||

20 | \(200\pi\) | |||

12 | \(64\pi\) | |||

3 | 3.14 |

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.

### 20.4: Popcorn Deals

A movie theater offers two containers:

Which container is the better value? Use 3.14 as an approximation for \(\pi\).

### Summary

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

\(\displaystyle V=\frac13 \pi r^2h\)

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 \(64\pi\) cm^{3}, a height of 3 cm, and an unknown radius \(r\). From the volume formula, we know that

\(\displaystyle 64 \pi = \frac{1}{3}\pi r^2 \boldcdot 3\)

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 \(18 \pi\) cm^{3}, a radius of 3 cm, and an unknown height \(h\). Using the formula for the volume of the cone, we know that

\(\displaystyle 18 \pi = \frac{1}{3} \pi 3^2h\)

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

### Glossary Entries

**base (of a prism or pyramid)**The word

*base*can also refer to a face of a polyhedron.A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

**cone**A cone is a three-dimensional figure like a pyramid, but the base is a circle.

**cross section**A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

**cylinder**A cylinder is a three-dimensional figure like a prism, but with bases that are circles.

**prism**A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

**pyramid**A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

**sphere**A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

**surface area**The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

For example, if the faces of a cube each have an area of 9 cm

^{2}, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm^{2}.**volume**Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units

^{3}, because it is composed of 3 layers that are each 20 units^{3}.