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# Surface Area and Volume of Cones

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Have you ever measured a platform that was rounded? Take a look at this dilemma.

The students ordered an award platform from a company for the Olympics. This platform would go on top of the cement area that the students already made and would be the place for the first place finisher in each event to stand and receive his/her award.

“It isn’t a trapezoid like the area we created,” Jose commented when the platform arrived at school.

“I know. They didn’t have a trapezoid. We got this instead,” Carmen explained.

“I think it will still work, and we can paint it,” Jose said.

“How much paint will we need?” Carmen asked.

“I’m not sure,” Jose said scratching his head.

This figure is a different kind of cone called a truncated cone. You can figure out the surface area of a figure like this one, but you will need a special formula. As you work through this Concept, look for the formula. You will have a chance to use it at the end of the Concept.

### Guidance

Cones are three-dimensional figures that have a base and a point at the top. However, cones always have a circular base.

We can calculate the surface area of a cone. Do you remember how to define surface area? Take a look.

Surface area is the total of the areas of each face in a solid figure.

Imagine that you could wrap a pyramid or cone in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, we must be able to calculate the area of each face and then add these areas together.

We will look at different ways to calculate surface area. One way is to use a net .

A net is a two-dimensional diagram of a three-dimensional figure.

Cones have different nets. Imagine you could unroll a cone.

Here is what the net of a cone would look like.

The shaded circle is the base. Remember, cones always have circular bases. The unshaded portion of the cone represents its side. Technically we don’t call this a face because it has a round edge.

To find the surface area of a cone, we need to calculate the area of the circular base and the side and add them together. The formula for finding the area of a circle is $A = \pi r^2$ , where $r$ is the radius of the circle. We use this formula to find the area of the circular base.

The side of the cone is actually a piece of a circle, called a sector. The size of the sector is determined by the ratio of the cone’s slant height to its radius, or $\frac{s}{r}$ .

To find the area of the cone’s side, we multiply the radius, the slant height, and pi.

$A=rs \pi$

Now, let's apply this information.

Find the surface area of the following cone.

Now that we have the measurements of the sides of the cone, let’s calculate the area of each. Remember to use the correct area formula.

$& \mathbf{Bottom \ face \ (circle)} && \mathbf{Side}\\& A = \pi r^2 && A = \pi rs\\& \pi (5^2) && \pi (5) (11.7)\\& 25 \pi && \pi (58.5)\\& 78.5 && 58.5 \pi$

We know the area of each side of the cone when we approximate pi as 3.14. Now we can add these together to find the surface area of the entire cone.

$&\text{bottom face} \qquad \qquad \text{side} \qquad \qquad \text{surface area}\\& \quad 78.5 \qquad \quad \ + \quad \ 183.69 \quad = \quad \ 262.19 \ in^2.$

We used the formula $A = \pi r^2$ to find the area of the circular base. Then we found the area of the side by multiplying $\pi rs$ .

When we add these together, we get a surface area of 262.19 square inches for this cone.

Here is the formula for finding the surface area of a cone.

$SA = \pi r^2 + \pi rs$

The first part of the formula, $\pi r^2$ , is simply the area formula for circles. This represents the base area. The second part, as we have seen, represents the area of the cone’s side. We simply put the pieces together and solve for the area of both parts at once.

Write this formula down in your notebook.

Now let’s look at how we can find the surface area of part of a cone.

First, let’s think about what a truncated cone would look like. A truncated cone is one where the point of the cone is cut off leaving two circular bases and a side face.

Here is what one looks like.

Notice that with a truncated cone, that we will have two different circular bases-a top radius and a bottom one. We will have to find the area of both bases plus the area of the sector to find the surface area.

The formula for finding the surface area of a truncated cone is:

$SA= \pi [s(R+r)+ R^2+r^2]$

Notice that s stands for slant height and the capital $R$ stands for the larger radius and the lowercase $r$ stands for the smaller radius.

What is the surface area of a truncated cone with a slant height of 6 cm, a radius of 8 cm and a radius of 6 cm?

To find the surface area of this figure, we fill the dimensions into the formula and solve.

$SA &= \pi[6(8+6)+8^2+6^2]\\SA &= \pi[84+64+36]\\SA &= 3.14(184)\\SA &= 577.76 \ cm^2$

Find the surface area of each cone.

#### Example A

A cone with a radius of 4 inches and a slant height of 6 inches.

Solution: $125.6$ sq. inches

#### Example B

A cone with a radius of 5 feet and a slant height of 8 feet.

Solution: $204.1$ sq. feet

#### Example C

A cone with a radius of 3 inches and a slant height of 4.5 inches.

Solution: $70.65$ sq. inches

Now let's go back to the dilemma from the beginning of the Concept.

We can use the following formula to find the surface area of a truncated cone.

$\pi [s(R+r)+ R^2+r^2]$

Now we can take the given information and substitute it into the formula.

$& 3.14[3(3 + 2) + 3^2 + 2^2]\\& 3.14 [3(5) + 9 + 4]\\& 3.14 [28]\\& SA = 87.92 \ ft^2$

### Vocabulary

Cone
a three – dimensional solid figure with a circular base, a sector for a side that meets in a single vertex.
Surface area
the measurement of the outer covering of a solid figure.
Net
a diagram that represents what a solid figure would look like in two-dimensions, flattened out as a pattern.
Sector
a piece of a circle.
Truncated cone
when a cone is divided by a plane so that the point is removed from the cone leaving only the bottom section.

### Guided Practice

Here is one for you to try on your own.

Trey is decorating conical party hats for his party by wrapping them in colored tissue paper. Each hat has a radius of 4.2 centimeters and a slant height of 8.6 centimeters. If he wants to wrap 6 party hats, how much paper will he need?

Solution

This problem involves a cone. It does not include a picture, so it may help to draw a net. In your drawing, label the radius and the slant height of the cone. We can also use the formula. We simply put the radius and slant height in for the appropriate variables in the formula and solve for $SA$ .

$SA &= \pi r^2 + \pi rs\\SA &= \pi (4.2^2) + \pi (4.2) (8.6)\\SA &= 17.64 \pi + 36.12 \pi\\ SA &= 53.76 \pi\\SA &= 168.81 \ cm^2$

Trey will need 168.81 square centimeters of tissue paper to cover one hat, when we approximate pi as 3.14.

But we’re not done yet! Remember, he wants to cover 6 party hats.

We need to multiply the surface area of one hat by 6 to find the total amount of paper he needs: $168.81 \times 6 = 1,012.86$ . Trey will need 1,012.86 square centimeters of paper to cover all 6 hats.

### Practice

1. What is the name of the figure represented in this net?
2. What is the diameter of this figure?
3. What is the slant height of the figure?
4. What is the surface area of the figure?

1. What is the name of this figure?
2. What is the shape of the base?
3. What is the diameter of the base?
4. What is the surface area of this figure?

1. What is the name of this figure?
2. What is the shape of the base?
3. What is the diameter of the base?
4. What is the surface area of this figure?

Directions: Find the surface area of each cone.

1. $r = 4 \ in, \ sh = 5 \ in$
2. $r = 5 \ m, \ sh = 7 \ m$
3. $r = 3 \ cm, \ sh = 6 \ cm$