# Lesson 19

The Volume of a Cone

Let’s explore cones and their volumes.

### 19.1: Which Has a Larger Volume?

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

- Which figure has a larger volume?
- Do you think the volume of the smaller one

is more or less than \(\frac12\) the volume of the

larger one? Explain your reasoning. - 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 \(h\).

Here is a method for quickly sketching a cone:

- 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.

### 19.2: From Cylinders to Cones

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

- If the volume of the cylinder is 90 cm
^{3}, what is the volume of the cone? - If the volume of the cone is 120 cm
^{3}, what is the volume of the cylinder? - 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.

### 19.3: Calculate That Cone

- 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 \(\pi\).

- Here is a cone.
- What is the area of the base? Express your answer in terms of \(\pi\).
- What is the volume of the cone? Express your answer in terms of \(\pi\).

- 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.

A grain silo has a cone shaped spout on the bottom in order to regulate the flow of grain out of the silo. The diameter of the silo is 8 feet. The height of the cylindrical part of the silo above the cone spout is 12 feet while the height of the entire silo is 16 feet.

How many cubic feet of grain are held in the cone spout of the silo? How many cubic feet of grain can the entire silo hold?

### 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.

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

### 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}.