 7.03 Circumference of a circle

Lesson

Circumference

The circumference of a circle is the perimeter of a circle. It can be calculated with a special formula.

Remember from our exploration, we have already learned about the special value pi, or $\pi$π, which will help us. We found that:

$\pi=\frac{C}{d}$π=Cd, where $C$C is the circumference and $d$d is the diameter

To find the formula for the circumference of a circle we will rearrange the ratio we used to find $\pi$π and solve for $C$C.

Circumference of a circle

$C=\pi d$C=πd

and because the diameter is twice the radius, we can also write the formula as

$C=2\pi r$C=2πr

Practice Questions

QUESTION 1

Find the circumference of the circle shown, correct to two decimal places. QUESTION 2

Find the circumference of the circle shown, correct to $2$2 decimal places. QUESTION 3

If the radius of a circle equals $27$27 cm, find its circumference correct to one decimal place.

Problem solving with circular shapes

If you take a look around, you should be able to spot some objects that have circular features. A clock face, a round plate, or the lid of a jar are some examples of things that have the outline of a circle. For lots of other common objects there are circles present in different parts of their shape. Some examples include the rim of a cup, or the crust on a pizza. Once we have found a feature of an object that looks like a circle, we can explore the relationship between the radius, the diameter, and the circumference of that circle. In his way we can understand more about the overall object.

Remember!

The diameter $d$d of a circle is equal to twice the radius $r$r, so $d=2r$d=2r.

If we know the diameter of a circle, the circumference $C$C is given by the equation $C=\pi d$C=πd.

If we know the radius of a circle, the circumference is given by the equation $C=2\pi r$C=2πr.

Worked example

Question 4

Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of $5$5 m. What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place.

Think: The distance around the outside of a circle is its circumference. We can use the radius of the pool to find its circumference.

Do: Using the equation $C=2\pi r$C=2πr, we substitute $r=5$r=5 m.

 $C$C $=$= $2\pi r$2πr Formula for circumference $=$= $2\pi\times5$2π×5 Substituting in $r=5$r=5 $=$= $10\pi$10π Simplifying partially $=$= $31.41592$31.41592$...$... Evaluating on a calculator $=$= $31.4$31.4 Rounding correctly

Lisa will walk $31.4$31.4 m around the pool.

Reflect: Suppose Lisa's neighbor has a circular pool with twice the circumference. What would be its radius?

Circular motion

We can also use these same ideas about circles to understand the circular motion of objects. Think about the way the end of the second hand moves around a clock, or the way a wheel rolls down a hill.

Worked example

Question 5

Harry pushes a hula hoop down a hill, and it rolls all the way to the bottom. The hula hoop has a diameter of $85$85 cm. If the hula hoop rolled a total of $25$25 m, how many revolutions did it make on its way down the hill? Give your answer to two decimal places.

Think: The hula hoop will make one full revolution when it rolls a distance equal to its circumference. We can use the diameter to find the circumference, then divide the total distance the hoop rolled by this length to find the number of revolutions.

We have both cm and m, so let's convert them to the same unit.

Do: If we work in m, the diameter of the hula hoop is $0.85$0.85 m.

 $C$C $=$= $\pi d$πd Formula for circumference $=$= $\pi\times0.85$π×0.85 Substitute in the diameter

Now we can divide the total distance rolled by this circumference.

 Number of revolutions $=$= $\frac{\text{total distance rolled }}{\text{circumference }}$total distance rolled circumference ​ $=$= $\frac{25}{0.85\pi}$250.85π​ $=$= $9.36$9.36 (to two d.p.)

The hula hoop has rolled about $9.36$9.36 times as it covers the distance of $25$25 m.

Reflect: How far would the hula hoop have rolled if it completed exactly $11$11 revolutions?

Practice questions

Question 6

A scooter tire has a diameter of $34$34 cm. What is the circumference of the tire?

Question 7

What is the length of the strip of  seaweed around the outside of the sushi? Question 8

Carl is performing an experiment by spinning a metal weight around on the end of a nylon thread. How far does the metal weight travel if it completes $40$40 revolutions on the end of a $0.65$0.65 m thread?