# 6.04 Volume and surface area of cones

Lesson

## Volume of a cone

The volume of a cone has the same relationship to a cylinder as we just saw that a pyramid has with a prism.

That is:

Volume of a right cone

$\text{Volume of Right Cone }=\frac{1}{3}\times\text{Area of Base }\times\text{Height of cylinder}$Volume of Right Cone =13×Area of Base ×Height of cylinder

$V=\frac{1}{3}\pi r^2h$V=13πr2h

Finding out the amount of snow cone that would fit inside one of those conical (cone-like) cups, you can use the volume equation to find the volume inside the cone, right?

#### Practice questions

##### QUESTIOn 1

Find the volume of the cone shown.

## Surface area of cones

### Cones

A cone is made by connecting a circular base to an apex.  If the apex is directly perpendicular to the center of the base, it is called a right cone. Cone shapes appear everywhere in the real world.

Using the interactive below you can see what happens when we unravel a cone.  This will help us to see the shapes we need to work out its surface area.

As we found with other 3D shapes, calculating surface areas is done by finding the total of the area of all faces.  For right cones, we have the base and a circle sector.

#### Exploration

We can see that the base is a circle, so it will have area $\pi r^2$πr2 where $r$r is the radius of the circular base of the cone.

The other piece is the cone portion. If it is opened up and flattened out it looks like a circle with a piece missing. This circle has a radius of $s$s, which is the slant height of the cone.

Before we work out the area of the cone portion, let's first consider the entire circle it is a part of.

The area of this large circle with radius $s$s, would be $\pi s^2$πs2

The circumference of the large circle with radius $s$s would be $2\pi s$2πs.

The pink arc, arc $AB$AB, originally wrapped around the base of the cone, and so its length is the circumference of the base. So the length of arc AB is $2\pi r$2πr

The ratio of the blue shaded sector to the area of the whole circle, is the same as the ratio of the pink arc AB to circumference of the whole circle.

We can write this as an equation.

 $\frac{\text{area of cone portion }}{\text{area of whole circle }}$area of cone portion area of whole circle ​ $=$= $\frac{\text{length of arc }}{\text{circumference of large circle }}$length of arc circumference of large circle ​ By definition of ratios $\frac{\text{area of cone portion}}{\pi s^2}$area of cone portionπs2​ $=$= $\frac{2\pi r}{2\pi s}$2πr2πs​ Substituting in given information $\text{area of sector }$area of sector $=$= $\frac{r}{s}\times\pi s^2$rs​×πs2 Simplifying and multiplying both sides by $\pi s^2$πs2 $\text{area of sector }$area of sector $=$= $\pi rs$πrs Simplifying

Thus the total surface area of a right cone is:

Surface area of right cone

Where $r$r is the cone's base radius and $l$l is the slant height:

 $\text{Surface Area of Right Cone}$Surface Area of Right Cone $=$= $\text{Area of Base }+\text{Area of Sector }$Area of Base +Area of Sector $SA$SA $=$= $\pi r^2+\pi rl$πr2+πrl

#### Practice questions

##### Question 3

Find the surface area of the cone shown.

##### Question 4

The top of a solid cone was sawed off to form the solid attached. Find the surface area of the solid formed correct to 2 decimal places.

## Practical problems involving cones

It is important that we are able to determine whether a question involves determining volume or surface area of a cone.  Once we have determined that the question will require us to deal with a cone, either by reading the text of a question or by viewing an image, we need to decide whether we will need to calculate the volume or the surface area of the cone.

Remember!
• Volume - the problem mentions filling the inside of the cone or uses cubic units
• Surface area - the problem mentions covering the outside of the cone or uses square units

#### Practice questions

##### question 5

A theater serves popcorn in a small conical container and a large cylindrical container as shown below.

1. How many small containers must be purchased in order to have the same amount of popcorn in the large container?

2. Find the volume of the smaller container in cubic centimeters.

3. Find the volume of the larger container in cubic centimeters.

4. The cost of a small container of popcorn is $\$1.30$$1.30 and the cost of a large container of popcorn is \3.50$$3.50. Which is the better buy?

Small container

A

Large container

B

Small container

A

Large container

B

##### question 6

Given the following right cone.  Use the pi button on your calculator for pi.

1. Find the surface area of the right cone. Give your answer correct to $2$2 decimal places.

2. Find the height of the right cone. Give your answer correct to $2$2 decimal places.

3. Find the volume of the right cone. Give your answer correct to $2$2 decimal places.

### Outcomes

#### 8.6a

Solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids