11.09 Circles, sectors and arc length

The circumference of a circle is the distance around the edge of a circle. In other words, 'circumference' is a specific term for the perimeter of the circle,C=2\pi r where C is the circumference and r is the radius of a circle.

For example, if the radius of a circle is 8\text{ cm}, the circumference, C=2\pi r=2\times \pi \times 8=50.3\text{ cm}(rounded to one decimal place).

As the arc of a circle is a fraction of the edge of a circle, we can calculate its arc length s using a variation of the circumference of a circle formula.

Imagine first that the arc length that we want is half of the edge of the circle, then we could take half the circumference and get s=\dfrac{1}{2} \times 2 \pi r.

What if we wanted a quarter of the edge? Then the arc length would be s=\dfrac{1}{4} \times 2 \pi r.

What if we want a different fraction? Particularly, the fraction created by using an angle \theta. Then we would use s=\dfrac{\theta}{360\degree}\,2\pi r.

If \theta is the angle at the centre of the circle, measured in degrees and subtended by an arc, then the arc length can be calculated using the formula: \begin{aligned}s&=\dfrac{\theta}{360\degree}\times 2\pi r \\ &= r \times \dfrac{\pi}{180\degree}\times \theta \end{aligned}

When finding the perimeter of a sector, don't forget to add the lengths of the straight edges to the arc length. \text{Perimeter of a Sector} = \text{Arc Length} + 2 r

Examples

Example 1

Find the length of the arc in the figure correct to one decimal place.

Worked Solution

Create a strategy

Use the formula for an arc: s=\dfrac{\theta}{360\degree} \times 2\pi r

Apply the idea

\displaystyle s	\displaystyle =	\displaystyle \dfrac{50\degree}{360\degree} \times 2 \pi \times 74	Substitute \theta=50,\,r=74
	\displaystyle =	\displaystyle 64.6\text{ km}	Simplify

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Example 2

If the arc formed by two points on a sphere with a radius of 2 \text{ m} subtends an angle of 37\degree at the centre, find the length of the arc correct to two decimal places.

Worked Solution

Create a strategy

Use the formula for an arc: s=\dfrac{\theta}{360\degree} \times 2\pi r

Apply the idea

\displaystyle \text{Arc length}	\displaystyle =	\displaystyle \dfrac{37\degree}{360\degree} \times 2 \pi \times 2	Substitute \theta=37,\,r=2
	\displaystyle =	\displaystyle 1.29 \text{ m}	Simplify

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The area of a full circle, measured in square units, can be found using the following formula: \text{Area of a circle}=\pi r^2

Given that the area of a circle is \pi r^2 thearea of a sector is some fraction of that full area \dfrac{\theta}{360\degree}\times \pi r^2.

There is a minor sector and a major sector associated with any given angle at the centre. The area of the corresponding sector can be found by replacing \theta with 360\degree-\theta in the formula for the area of a sector, to become \dfrac{360\degree-\theta}{360\degree}\times \pi r^2. Notice this simplifies to \pi r^2-\dfrac{\theta }{360}\times \pi r^2, i.e. subtracting the area of the original sector from the area of the whole circle.

We can make a formula for the area of any sector depending on the angle \theta that subtends the arc at the centre.

A sector with contained angle \theta corresponds to a fraction \dfrac{\theta}{360\degree} of a full circle and so its area is given by: A = \dfrac{\theta}{360\degree} \times \pi r^2

Examples

Example 3

Find the area of the following sector of a circle. Round your answer to one decimal place.

Worked Solution

Create a strategy

Use the formula for the area of sector:\text{ Area}=\dfrac{\theta}{360\degree}\times\pi r^{2}

Apply the idea

\displaystyle \text{Area}	\displaystyle =	\displaystyle \dfrac{121\degree}{360\degree}\times \pi \times 13^{2}	Substitute r=13 and \theta =121
	\displaystyle =	\displaystyle 178.5\text{ cm}^{2}	Evaluate and round

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Example 4

Consider the sector below.

a

Calculate the perimeter. Give your answer correct to one decimal place.

Worked Solution

Create a strategy

Add the two radii and the arc length.

Apply the idea

We write the equation as \text{Perimeter}=2r + \dfrac{\theta}{360\degree}\times 2\pi r,

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2\times 96.4 + \dfrac{90\degree}{360\degree}\times 2\pi \times 96.4	Substiute r=96.4,\,\theta =90
	\displaystyle =	\displaystyle 344.2 \text{ mm}	Evaluate

b

Calculate the area. Give your answer correct to four decimal places.

Worked Solution

Create a strategy

Use the formula for the area of sector:\text{ Area}=\dfrac{\theta}{360\degree}\times\pi r^{2}

Apply the idea

\displaystyle \text{Area}	\displaystyle =	\displaystyle \dfrac{90\degree}{360\degree}\times \pi \times 96.4^{2}	Substitute r=96.4 and \theta =90
	\displaystyle =	\displaystyle 7298.6737\text{ mm}^{2}	Evaluate and round