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# 8.7: Cone: Base Area, Lateral Area, Surface Area and Volume

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Find the surface area of a cone using a net or a formula.
• Find the volume of a cone.

## Cones

A cone is a three-dimensional figure with a single curved base that tapers to a single point called an apex. The base of a cone can be a circle or an oval of some type. In this chapter, we will only use circular cones.

• The ______________________ is the point on top of a cone.
• The base of the cones we will study is in the shape of a ____________________.

You can remember the name “cone” of this shape because it looks like an upside-down ice cream cone.

The apex of a right cone lies above the center of the cone’s circle. In an oblique cone, the apex is not in the center:

• The apex of a right cone is the point directly above the _____________________ of the cone’s circular base.

The height of a cone $h$ is the perpendicular distance from the center of the cone’s base to its apex.

• The height of a cone is just like an altitude: a _______________________ line from the center of the circular base to the apex.

## Surface Area of a Cone Using Nets

Most three-dimensional figures are easy to deconstruct into a net. The cone is different in this regard. Can you predict what the net for a cone looks like? In fact, the net for a cone looks like a small circle and a sector, or part of a larger circle.

The diagram below shows how the half-circle sector folds to become a cone:

Note that the circle that the sector is cut from is much larger than the base of the cone.

• The net for a cone is a circular base plus a _____________________.

Example 1

Which sector will give you a taller cone—a half circle or a sector that covers three-quarters of a circle? Assume that both sectors are cut from congruent circles.

Make a model of each sector:

1. The half circle makes a cone that has a height that is about equal to the radius of the semi-circle.

2. The three-quarters sector gives a cone that has a wider base (greater diameter) but its height as not as great as the half-circle cone.

Example 2

Predict which will be greater in height—a cone made from a half-circle sector or a cone made from a one-third-circle sector. Again, assume that both sectors are cut from congruent circles.

The relationship in Example #1 on the previous page holds true—the greater (in degrees) the sector, the smaller the height of the cone.

In other words, the fraction $\frac{1}{3}$ is less than $\frac{1}{2}$, so a one-third sector will create a cone with greater height than a one-half sector.

• The larger the sector, the __________________________ the height of its cone.

Example 3

Predict which will be greater in diameter—a cone made from a half-circle sector or a cone made from a one-third-circle sector. Assume that the sectors are cut from congruent circles.

Here you have the opposite relationship—the larger (in degrees) the sector, the greater the diameter of the cone.

In other words, $\frac{1}{2}$ is greater than $\frac{1}{3}$, so a one-half sector will create a cone with greater diameter than a one-third sector.

• The larger the sector, the ________________________ the diameter of its cone.

## Surface Area of a Regular Cone

The surface area of a regular pyramid is given by:

$A = \left ( \frac{1}{2} l P \right ) + B$

where $l$ is the slant height of the figure, $P$ is the perimeter of the base, and $B$ is the area of the base.

Imagine a series of pyramids in which $n$, the number of sides of each figure’s base, increases.

As you can see, as $n$ increases, the figure more and more resembles a circle.

You can also think of this as: a circle is like a polygon with an infinite number of sides that are infinitely small.

Similarly, a cone is like a pyramid that has an infinite number of sides that are infinitely small in length.

As a result, the formula for finding the total surface area of a cone is similar to the pyramid formula. The only difference between the two is that the pyramid uses $P$, the perimeter of the base, while a cone uses $C$, the circumference of the base.

$A ( \text{pyramid} ) = \frac{1}{2} l P + B && and && A ( \text{cone}) = \frac{1}{2}l C + B$

Since the circumference of a circle is $2 \pi r$:

$A ( \text{cone} ) = \frac{1}{2} l C + B = \frac{1}{2} l (2 \pi r) + B = \pi rl + B$

You can also express $B$ as $\pi r^2$ to get:

## Surface Area of a Right Cone

$A ( \text{cone} ) &= \pi rl + B\\&= \pi rl + \pi r^2\\&= \pi r ( l + r )$

Any of these forms of the equation can be used to find the surface area of a right cone.

There are a few different formulas to find the surface area of a cone. Pick one formula and describe what every variable represents.

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Example 4

Find the total surface area of a right cone with a radius of 8 cm and a slant height of 10 cm.

Use the formula:

$A ( \text{cone} ) & = \pi r ( l + r )\\& = \pi(8) \cdot (10 + 8)\\& = 8 \pi \cdot 18\\& = 144 \pi \ cm^2 \quad \text{or}\\& \approx (144)(3.14) = 452.16 \ cm^2$

The exact area of the cone is $144 \pi \ cm^2$ and the approximate area is $452.16 \ cm^2$.

Example 5

Find the total surface area of a right cone with a radius of 3 feet and an altitude (not slant height) of 6 feet.

Use the Pythagorean Theorem to find the slant height $l$:

$r^2 + h^2 &= l^2\\3^2 + 6^2 &= l^2\\9 + 36 &= l^2\\45 &= l^2\\\sqrt{45} &= l\\3 \sqrt{5} &= l$

Now use the area formula:

$A ( \text{cone} ) &= \pi r ( l + r )\\&= \pi (3) \cdot ( 3 \sqrt{5} + 3)\\&= 3 \pi ( 3 \sqrt{5} + 3)$

If we leave this as an exact answer, we cannot simplify anymore. This would be an ideal time to use a decimal approximation with a calculator:

$3 \pi ( 3 \sqrt{5} + 3) \approx 3(3.14)( 3 \sqrt{5} + 3) = 91.45 \ cm^2$

The surface area of the cone is approximately $91.45 \ cm^2$.

## Volume of a Cone

Which has a greater volume, a pyramid, cone, or cylinder if the figures have bases with the same "diameter" (i.e., distance across the base) and the same altitude?

To find out, compare pyramids, cylinders, and cones that have bases with equal diameters and the same altitude.

Here are three figures that have the same dimensions—cylinder, a right regular hexagonal pyramid, and a right circular cone. Which figure appears to have a greater volume?

It seems obvious that the volume of the cylinder is greater than the other two figures. This is because the pyramid and cone taper off to a single point, while the cylinder’s sides stay the same width.

Determining whether the pyramid or the cone has a greater volume is not so obvious. If you look at the bases of each figure you see that the apothem of the hexagon is congruent to the radius of the circle. You can see the relative size of the two bases by superimposing one onto the other:

From the diagram you can see that the hexagon is slightly larger in area than the circle.

Therefore, the volume of the right hexagonal regular pyramid would be greater than the volume of a right circular cone. It is, but only because the area of the base of the hexagon is slightly greater than the area of the base of the circular cone.

• When comparing the volumes of a cylinder, a pyramid, and a cone, the __________________________ has the largest volume and the __________________________ has the smallest volume. The __________________________ has a volume in between the other two shapes.

The formula for finding the volume of each figure is virtually identical. Both formulas follow the same basic form:

$V = \frac{1}{3} Bh$

Since the base of a circular cone is, by definition, a circle, you can substitute the area of a circle, $\pi r^2$ for the base of the figure. This is expressed as a volume postulate for cones.

• Instead of using $B$ for base area, we use the area of a _____________________, $\pi r^2$, in the formula for volume of a cone.

Volume of a Right Circular Cone

Given a right circular cone with height $h$ and a base that has radius $r$:

$V &= \frac{1}{3} Bh\\V &= \frac{1}{3} \pi r^2 h$

Example 6

Find the volume of a right cone with a radius of 9 cm and a height of 16 cm.

Use the formula: $V = \frac{1}{3} \pi r^2 h$

Substitute the values for $r =$ ___________ and $h =$ _____________ :

$V & = \frac{1}{3} \pi (9^2)(16)\\V & = \frac{1296 \pi}{3} = 432 \pi \ cm^3 \quad \text{or}\\& \approx (432)(3.14) = 1356.48 \ cm^3$

The cone has an exact volume of $432 \pi$ cubic centimeters and an approximate volume of 1356.48 cubic centimeters.

By now, you have seen the units $cm^2$ or $in^2$ and $cm^3$ or $in^3$ in the examples.

When we calculate area, we use a “square” unit, such as $cm^2$ (square centimeters) or $in^2$ (square inches)

When we calculate volume, we use a “cubic” unit, such as $cm^3$ (cubic centimeters) or $in^3$ (cubic inches)

Example 7

Find the volume of a right cone with a radius of 10 feet and a slant height of 13 feet.

Use the Pythagorean theorem to find the height $h$:

$r^2 + h^2 & =l2\\10^2 + h^2 & =132\\100 + h^2 & =169\\h^2 & = 169 - 100 =69\\h & = \sqrt{69} \approx 8.31 \ ft$

Now use the volume formula: $V = \frac{1}{3} \pi r^2 h$

Substitute the values for $r =$ ___________ and $h =$ _____________ :

$V &= \frac{1}{3} \pi (10^2)(8.31)\\V & = \frac{831 \pi}{3} = 277 \pi \ ft^3 \quad \text{or}\\& \approx (277)(3.14) = 869.78 \ ft^3$

The cone’s volume can be written as $277 \pi \ ft^3$ or $869.78 \ ft^3$.

1. What type of units are used to express volume? What type for area?

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2. What shape is the base of a right circular cone?

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3. When calculating the volume of a cone, what information do you need?

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## Date Created:

Feb 23, 2012

May 12, 2014
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