Geometry (B): READ: Surface Area of a Cone

READ: Surface Area of a Cone

Cones

Learning Objectives

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

Introduction

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 limit the discussion to circular cones. 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 height of a cone, $h$ , is the perpendicular distance from the center of the cone’s base to its 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 diagrams below show 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.

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.

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

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. Assume that both sectors are cut from congruent circles.

The relationship in the example above holds true—the greater (in degrees) the sector, the smaller in height of the cone. In other words, the fraction $1/3$ is less than $1/2$ , so a one-third sector will create a cone with greater height than a half sector.

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, $1/2$ is greater than $1/3$ , so a one-half sector will create a cone with greater diameter than a one-third sector.

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. So in a sense, a circle approaches a polygon with an infinite number of sides that are infinitely small.

In the same way, a cone is like a pyramid that has an infinite number of sides that are infinitely small in length.

Given this idea, it should come as no surprise that 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.

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

Surface Area of a Right Cone

The surface area of a right cone is given by:

$A = \frac{1}{2} l C + B$

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

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

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

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

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

Example 4

Find the total surface area of a right cone with a radius of $8 \;\mathrm{cm}$ and a slant height of $10\;\mathrm{cm}$ .

Use the formula:

$A \text{(cone)} &= \pi r (l + r)\\ &= (3.14)(8)[10 + 8]\\ &= 452.16 \ \text{square cm}$

Example 5

Find the total surface area of a right cone with a radius of $3\;\mathrm{feet}$ and an altitude (not slant height) of $6\;\mathrm{feet}$ .

Use the Pythagorean Theorem to find the slant height:

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

Now use the area formula.

$A \text{(cone)} &= \pi r (l + r)\\ &= (3.14)(3)[3\sqrt{5} + 3]\\ &\approx 91.5 \ \text{square cm}$

Last modified: Tuesday, 6 July 2010, 10:01 AM

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