# 5.02 Surface area of cylinders and spheres

Lesson

### Cylinders

Although a cylinder is not a prism - because it has a curved surface between the ends - just like a prism we can use its net to help us develop a rule for finding its surface area.

Let's see how it would look if we open up a cylinder to view its net.

When the net is unfolded, the curved surface is shown to be a rectangle. We can see that there are three parts to a cylinder's surface area - two circles and a rectangle.

 Surface area of a cylinder $=$= area of $2$2 circular ends  + area of $1$1 rectangular piece $=$= $\left(2\times\pi r^2\right)+\left(L\times W\right)$(2×πr2)+(L×W)

We know that the length of the rectangle is the height, $h$h, of the cylinder. By rotating the circle on top of the rectangle, we can see how the circumference of the circle is equal to the width of the rectangular piece? The circumference of a the circle is given by $2\pi r$2πr, so we have:

Surface Area of a Cylinder

$A=2\pi r^2+2\pi rh$A=2πr2+2πrh

Questions can involve fractions of cylinders: for example, a water trough might be made from exactly half of a cylinder. We might be told that a cylinder is open, which would mean it has no circles as part of its surface area. Or, it might have one end closed, such as a water tank collecting rain. So it is important that we understand exactly how the formula above works so that we can use it in these more complicated scenarios.

#### Worked Example

##### Example 1

Calculate the surface area of the closed half cylinder below, giving your answer to $1$1 decimal place.

Think: The surface area is composed of half of a closed cylinder and an extra rectangle on top.

Do: The surface area of a cylinder =  $2\pi r^2+2\pi rh$2πr2+2πrh, so

The surface area of half a cylinder = $\pi r^2+\pi rh$πr2+πrh $=\pi\times3.6^2+\pi\times10\times3.6=153.8123$=π×3.62+π×10×3.6=153.8123...

The rectangle has its length equal to the height of the cylinder, and its width is equal to the diameter of the cylinder's ends.

We have the area of the rectangle $=10\times7.2=72$=10×7.2=72

So the total surface area is $153.8123$153.8123... $+72=225.8$+72=225.8 square units ($1$1 d.p.)

#### Practice Questions

##### QUESTION 1

Consider the following cylinder with a height of $35$35 cm and base radius of $10$10 cm. Find the surface area of the cylinder.

##### QUESTION 2

A cylindrical can of radius $7$7 cm and height $10$10 cm is open at one end. What is the external surface area of the can correct to two decimal places?

##### Question 3

Consider the cylinder shown in the diagram below.

1. Find the surface area of the cylinder in square centimetres.

2. Use your answer from part (a) to find the surface area of the cylinder in square millimetres?

### Spheres

A solid 3-dimensional circular object is a sphere.  Its surface is defined as the collection of points that are all equidistant from a central point (centre of the sphere).  Half of a sphere is called a hemisphere.

Unlike solids we have seen so far, we cannot unwrap a sphere to get a 2D net and calculate its area. But this didn't stop mathematicians finding a way! Archimedes investigated spheres relative to cylinders, and eventually determined the formula for the surface area of a sphere to be:

Surface Area of a Sphere

$A=4\pi r^2$A=4πr2

Just as with cylinders, we might be asked to work with hemispheres or other fractions of spheres.

#### Practice Questions

##### Question 4

Find the surface area of the sphere shown.

##### Question 5

Consider the following hemisphere with a radius of $8$8. Find the total surface area.

##### Question 6

A ball has a surface area of $A=50.27$A=50.27 mm2. What is its radius?