 # 4.03 Circumference of circles and sectors

Lesson

## Circumference

The perimeter of a circle is called the circumference, and can be found using the radius or the diameter. Remember that the radius is the distance from the centre of the circle to the circumference, and the diameter is twice the radius. Circumference of a circle

To calculate the circumference using the radius, $r$r:

 Circumference $=$= $2\times\text{pi }\times\text{radius }$2×pi ×radius $C$C $=$= $2\times\pi\times r$2×π×r $C$C $=$= $2\pi r$2πr

To calculate the circumference using the diameter, $D$D:

 Circumference $=$= $\text{pi }\times\text{diameter }$pi ×diameter $C$C $=$= $\pi\times D$π×D $C$C $=$= $\pi D$πD

#### Practice questions

##### QUESTION 1

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

##### QUESTION 2

Find the circumference of the circle shown, correct to three decimal places. ### Finding radius or diameter, given the circumference

Sometimes we know the circumference of a circle, but need to determine the radius or diameter.

We can rearrange the formula for the circumference, $C=2\pi r$C=2πr, which uses radius, to make radius the subject:

 $C$C $=$= $2\pi r$2πr $\frac{C}{2\pi}$C2π​ $=$= $\frac{2\pi r}{2\pi}$2πr2π​ (Divide both sides by $2\pi$2π) $\frac{C}{2\pi}$C2π​ $=$= $r$r (Simplify) $r$r $=$= $\frac{C}{2\pi}$C2π​

In a similar way, we can rearrange the formula for the circumference, $C=\pi D$C=πD, which uses diameter, to make diameter the subject:

 $C$C $=$= $\pi D$πD $\frac{C}{\pi}$Cπ​ $=$= $\frac{\pi D}{\pi}$πDπ​ (Divide both sides by $\pi$π) $\frac{C}{\pi}$Cπ​ $=$= $D$D (Simplify) $D$D $=$= $\frac{C}{\pi}$Cπ​

If we know the circumference $C$C of a circle, we can find its radius, $r$r, using the relation:

 $r$r $=$= $\frac{C}{2\pi}$C2π​
Diameter of a circle

If we know the circumference $C$C of a circle, we can find its diameter, $D$D, using the relation:

 $D$D $=$= $\frac{C}{\pi}$Cπ​

#### Practice questions

##### Question 3

What is the radius $r$r of a circle with the circumference $C=14$C=14 cm?

##### Question 4

Consider the circle below.

What is the diameter $D$D of the circle? ## Arc lengths and sectors

An arc of a circle is a curved line formed from part of the circumference of the circle. The length of an arc is called the arc length, $l$l.

sector is a shaped formed from part of a circle, where the sector's boundary or perimeter is formed by two radii and an arc.

Finding the perimeter of a sector involves first calculating the arc length, then adding on the lengths of the two radii.   Examples of sectors

Note that a quarter of a circle, is sometimes called a quadrant, and half of a circle is usually called a semicircle.

#### Worked example

##### Example 1

Find the perimeter of the sector equal to $\frac{1}{4}$14 of a circle with radius $6.5$6.5 cm.

Think: To find the perimeter of the sector, we start by finding the arc length, $l$l. This is equal to $\frac{1}{4}$14 of the circumference of the circle.

Do:

 arc length $=$= $\frac{1}{4}\times\text{circumference }$14​×circumference $l$l $=$= $\frac{1}{4}\times2\pi r$14​×2πr $=$= $\frac{1}{4}\times2\times\pi\times6.5$14​×2×π×6.5 (Substitute $r=6.5$r=6.5) $=$= $3.25\pi$3.25π (Leave answer in terms of $\pi$π)

If we leave our answer for arc length in terms of pi, we eliminate any rounding error in our calculations.

To get the perimeter, we must add the lengths of the two radii to the arc length.

 perimeter $=$= $\text{arc length }+2\times\text{radius }$arc length +2×radius $=$= $3.25\pi+2\times6.5$3.25π+2×6.5 (Substitute $r=6.5$r=6.5) $=$= $3.25\pi+13$3.25π+13 $=$= $23.2101$23.2101... $=$= $23.2$23.2 cm ($1$1 d.p.) (Rounded to $1$1 decimal place)

#### Practice question

##### Question 5

What is the perimeter of a semicircle with diameter $8$8 cm, correct to two decimal places?

## Sectors using degrees

Sectors are not always expressed as simple fractions of a circle. More generally they are expressed in terms of degrees. We can use the fact that a full circle is made up of $360^\circ$360° to calculate exactly what fraction of a circle is equal to the sector. We know already that a sector with a right angle ($90^\circ$90°) is quarter of a circle, because $\frac{90}{360}=\frac{1}{4}$90360=14.

For a sector with angle, $\theta$θ, between the two radii, we can calculate what fraction it is of a circle, by dividing the angle, $\theta$θ, by $360^\circ$360°.

Arc length and perimeter of a sector formed by angle $\theta$θ

To calculate arc length, $l$l:

 arc length $=$= $\frac{\theta}{360^\circ}\times\text{circumference of circle }$θ360°​×circumference of circle $l$l $=$= $\frac{\theta}{360^\circ}\times2\pi r$θ360°​×2πr

To calculate the perimeter of a sector:

 Perimeter of a sector $=$= $\text{arc length }+\left(2\times\text{radius }\right)$arc length +(2×radius )

#### Practice question

##### Question 6

Find the perimeter of the sector shown, correct to two decimal places. ## Problem solving with circular shapes

All around us there are objects with 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 common objects, circles are 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

##### Example 2

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.

Now we have $C=2\pi\times5=31.41592$C=2π×5=31.41592$...$... m, which rounds to $31.4$31.4 m. So Lisa will walk $31.4$31.4 m around the pool.

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

#### Practice questions

##### Question 7

A scooter tyre has a diameter of $34$34 cm.

Determine the circumference of the tyre, correct to one decimal place.

##### Question 8

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