Geometry (B): READ: Surface Area of Cylinders

READ: Surface Area of Cylinders

Cylinders

Learning Objectives

Find the surface area of cylinders.
Find the volume of cylinders.
Find the volume of composite three-dimensional figures.

Introduction

A cylinder is a three-dimensional figure with a pair of parallel and congruent circular ends, or bases. A cylinder has a single curved side that forms a rectangle when laid out flat.

As with prisms, cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular bases. The side of an oblique cylinder is not perpendicular to its bases.

Surface Area of a Cylinder Using Nets

You can deconstruct a cylinder into a net.

The area of each base is given by the area of a circle:

$A &= \pi r^2 \\ &=\pi (5)^2 \\ &=25 \pi \\ &\approx (25)(3.14) = 78.5$

The area of the rectangular lateral area $L$ is given by the product of a width and height. The height is given as $24$ . You can see that the width of the area is equal to the circumference of the circular base.

To find the width, imagine taking a can-like cylinder apart with a scissors. When you cut the lateral area, you see that it is equal to the circumference of the can’s top. The circumference of a circle is given by $C = 2 \pi r,$

the lateral area, $L$ , is

$L &= 2 \pi r h \\ &=2 \pi (5) (24)\\ &=240 \pi \\ &\approx (240)(3.14) = 753.6$

Now we can find the area of the entire cylinder using $A =\text{(area of two bases)} + \text{(area of lateral side)}$ .

$A &= 2 (75.36) + 753.6\\ &= 904.32$

You can see that the formula we used to find the total surface area can be used for any right cylinder.

Area of a Right Cylinder

The surface area of a right cylinder, with radius $r$ and height $h$ is given by $A=2 B +L$ , where $B$ is the area of each base of the cylinder and $L$ is the lateral area of the cylinder.

Example 1

Use a net to find the surface area of the cylinder.

First draw and label a net for the figure.

Calculate the area of each base.

$A &= \pi r^2 \\ &=\pi (8)^2 \\ &=64 \pi \\ &\approx (64)(3.14) = 200.96$

Calculate $L$ .

$L &= 2 \pi r h \\ &=2 \pi (8) (9)\\ &=144 \pi \\ &\approx (240)(3.14) = 452.16$

Find the area of the entire cylinder.

$A &= 2 (200.96) + 452.16\\ &= 854.08$

Thus, the total surface area is approximately $854.08\;\mathrm{square\ units}$

Surface Area of a Cylinder Using a Formula

You have seen how to use nets to find the total surface area of a cylinder. The postulate can be broken down to create a general formula for all right cylinders.

$A = 2B + L$

Notice that the base, $B$ , of any cylinder is:

$B= \pi r^2$

The lateral area, $L$ , for any cylinder is:

$L &= \text{width of lateral area}\cdot \text{height of cylinder}\\ &= \text{circumference of base}\cdot \text{height of cylinder}\\ &= 2\pi r \cdot h$