4.9 - Arc lengths and sector areas

Use a central angle as a fraction of a full turn to find an arc length or sector area. You will include the straight edges in a sector's perimeter and work backwards to find an unknown angle or radius, using degrees throughout.

Arc length from an angle (Higher)

A fan-shaped sheet has radius $6\,\mathrm{cm}$ and central angle $80^\circ$ . Its curved edge is $8\pi/3\,\mathrm{cm}$ long. Is that enough ribbon to edge the whole sheet? We will trace its complete boundary after finding out how the angle controls the curve.

You already know

A full turn is $360^\circ$ . A circle has circumference $2\pi r$ and area $\pi r^2$ . Both circle formulae are absent from either tier's sheet.

An arc is part of a circle's circumference. Equal central angles cut equal arc lengths from the same circle. A $90^\circ$ angle selects one quarter of a circumference; the same fraction idea works for other angles.

Sector

A sector is a circular region bounded by two radii and the arc between them. A region bounded by an arc and a chord instead is not a sector.

For $0<\theta<360$ , the fraction of a full circle is $\theta/360$ . A major sector uses a reflex angle greater than $180^\circ$ ; a minor sector uses the smaller angle. Read which region is named rather than automatically choosing the smaller one.

[DIAGRAM: asset_slug: m076_sector_boundary; caption: A sector with two radii and an 80 degree arc; description: OA and OB are equal radii of 6 centimetres, angle AOB is 80 degrees, and arc AB is the curved edge. The two radial edges also belong to the whole boundary; file: m076_sector_boundary.png; recommended_method: matplotlib]

The angle fraction multiplies the full circumference to give the curved length.

Arc length in degrees

L=\frac{\theta}{360}\times2\pi r

Here $r>0$ and $\theta$ is the sector angle in degrees. Arc length and sector formulae are absent from both tier sheets; this sector content is Higher. Not in the formulae sheet: learn it.

Start with the curve on the opening sheet.

Worked example 1: the fan's curved edge

Question. A sector has radius $6\,\mathrm{cm}$ and angle $80^\circ$ . Find its arc length exactly.

Plan. Multiply the full circumference by the fraction $80/360$ .

\begin{aligned} \frac{\theta}{360}&=\frac{80}{360}=\frac29 &&\text{fraction of a full turn}\\ L&=\frac29\times2\pi\times6 &&\text{same fraction of circumference}\\ &=\frac{8\pi}{3}\,\mathrm{cm} &&\text{keep the answer exact} \end{aligned}

Why use circumference? The requested measurement follows a curved boundary, so it is a length.

Answer. $L=8\pi/3\,\mathrm{cm}$ .

Check. $L/(12\pi)=2/9$ , the same fraction as $80/360$ .

Change only the angle from $80^\circ$ to $280^\circ$ . This names the remaining major sector of the same circle.

Worked example 2: the long arc

Question. A sector has radius $6\,\mathrm{cm}$ and angle $280^\circ$ . Find its arc length exactly.

Plan. Use the reflex angle to select the long arc.

\begin{aligned} \frac{\theta}{360}&=\frac{280}{360}=\frac79 &&\text{major-sector fraction}\\ L&=\frac79\times12\pi &&\text{fraction of circumference}\\ &=\frac{28\pi}{3}\,\mathrm{cm} &&\text{simplify} \end{aligned}

Why use $280$ ? The question asks for the major sector, so the smaller $80^\circ$ angle selects the wrong arc.

Answer. $L=28\pi/3\,\mathrm{cm}$ .

Check. $28\pi/3+8\pi/3=12\pi\,\mathrm{cm}$ , the full circumference.

The fraction must be between zero and one for a sector smaller than a full circle.

Watch out

$L=(360/80)\times12\pi$ reverses the fraction and produces more than a full circumference. Correct it to $L=(80/360)\times12\pi$ . Keep this fraction exact until the end.

All angles here use degrees. Radian measure is not needed for this course.

Sector area (Higher)

A sector's region is the same fraction of the circle as its arc is of the circumference. The fraction stays the same, but the quantity it multiplies changes.

Sector area in degrees

A=\frac{\theta}{360}\times\pi r^2

Use the angle of the required region in degrees and a positive radius. This formula is absent from both tier sheets; sector area is Higher content. Not in the formulae sheet: learn it.

Equal central angles divide a circle into equal regions. That is why the angle fraction applies to area as well as arc length. The result now uses square units.

Worked example 3: area of a major sector

Question. A major sector has radius $5\,\mathrm{cm}$ and angle $288^\circ$ . Find its area exactly.

Plan. Find the full circle area and take the fraction $288/360$ .

\begin{aligned} A_{\text{circle}}&=\pi\times5^2=25\pi\,\mathrm{cm}^2 &&\text{square the radius}\\ A&=\frac{288}{360}\times25\pi &&\text{required angle fraction}\\ &=\frac45\times25\pi=20\pi\,\mathrm{cm}^2 &&\text{simplify} \end{aligned}

Why is this more than half the circle? The selected angle $288^\circ$ is more than $180^\circ$ .

Answer. $A=20\pi\,\mathrm{cm}^2$ .

Check. The missing fraction is $1/5$ , whose area is $5\pi$ . The two areas sum to $25\pi\,\mathrm{cm}^2$ .

Change only the angle to the remaining $72^\circ$ , keeping the radius $5\,\mathrm{cm}$ .

Worked example 4: the remaining region

Question. A sector has radius $5\,\mathrm{cm}$ and angle $72^\circ$ . Find its area exactly.

Plan. Use the smaller fraction of the same full circle area.

\begin{aligned} A&=\frac{72}{360}\times\pi\times5^2 &&\text{angle fraction of area}\\ &=\frac15\times25\pi &&\text{simplify the fraction}\\ &=5\pi\,\mathrm{cm}^2 \end{aligned}

Why has only the fraction changed? The circle's radius, and therefore its full area, stay fixed.

Answer. $A=5\pi\,\mathrm{cm}^2$ .

Check. $72+288=360$ and $5\pi+20\pi=25\pi$ , so the two sectors complete the circle.

Here is a useful surprise: radius $10\,\mathrm{cm}$ with angle $72^\circ$ has area $(72/360)\pi\times10^2=20\pi\,\mathrm{cm}^2$ , equal to the major sector in example $3$ . Doubling a radius multiplies area by four, while quartering the angle divides it by four. Very different sectors can cover the same area.

Watch out

$A=(\theta/360)\times2\pi r$ uses the arc-length formula, so its result is a length. For area, use $A=(\theta/360)\times\pi r^2$ and square units.

Check that the selected angle describes the shaded or named region; the diagram may show the other angle more prominently.

Sector perimeter (Higher)

Trace from the centre along one radius, around the arc, and back along the other radius. A sector has three boundary pieces, even when its angle is reflex.

Sector perimeter

P=L+2r=\frac{\theta}{360}\times2\pi r+2r

This applies when both radial edges are exposed. It is absent from both tier sheets; sector perimeter is Higher content. Not in the formulae sheet: learn it.

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Number

Algebra

Shape, space and measure

Handling data

4.9 - Arc lengths and sector areas

Arc length from an angle (Higher)

Sector area (Higher)

Sector perimeter (Higher)

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