C5.3; E5.3 - Arcs and sectors

C5.3; E5.3 - Arcs and sectors

A slice of a circle has both a curved edge and two straight edges. You will use equal shares of a full turn to find arc lengths and sector areas, then decide which lengths belong in the sector's total perimeter.

A sector as a fraction of a circle

Take one sixth of a circular sheet. Is its perimeter one sixth of the sheet's circumference? Its curved edge is, but cutting the slice creates two straight edges. To count these correctly, first decide what fraction of the circle you have.

Sector and arc

A sector is the region bounded by two radii and the arc joining their endpoints. An arc is a portion of the circle's curved boundary. The sector angle is the angle between the two radii, measured through the sector.

A semicircle is a sector with angle 180∘180^\circ. A quarter-circle sector has angle 90∘90^\circ.

A full turn is 360∘360^\circ. Six equal 60∘60^\circ angles fill it, so each corresponding sector is one sixth of the circle. Equal angles at the centre cut off equal arc lengths and equal areas in the same circle. This lets you use fractions of the whole, without measuring a curved line.

[DIAGRAM: asset_name: Equal slices of a circle; asset_slug: equal_circle_slices; caption: A full turn of 360 degrees fixes the fraction represented by one slice.; description: One exact circle is divided into six 60 degree sectors with one shaded, and a second into three 120 degree sectors with one shaded. Each shaded sector has its angle marked at the centre and the captions identify one sixth and one third.]
A full turn of 360 degrees fixes the fraction represented by one slice.

Here we use angles that are factors of 360360: the angle fits a whole number of times around the centre. Examples include 60∘60^\circ, 90∘90^\circ, 120∘120^\circ and 180∘180^\circ. For a positive sector angle θ\theta below 360∘360^\circ, its fraction is θ/360∘\theta/360^\circ, with numerator and denominator both in degrees.

Worked example 1: one of six equal sectors

Question. What fraction of a circle is a 60∘60^\circ sector?

Plan. Count how many 60∘60^\circ angles make a full turn.

360÷60=6equal sectors in the wholefraction=16one of those six60360=16equivalent angle ratio\begin{aligned} 360\div60&=6 && \text{equal sectors in the whole}\\ \text{fraction}&=\frac16 && \text{one of those six}\\ \frac{60}{360}&=\frac16 && \text{equivalent angle ratio} \end{aligned}

Answer. 16\frac16 of the circle.

Check. Six copies have total angle 6×60∘=360∘6\times60^\circ=360^\circ, and their fractions add to 6×16=16\times\frac16=1.

Change only the angle to twice its previous size. The share of the circle should double too.

Worked example 2: a wider sector

Question. What fraction of a circle is a 120∘120^\circ sector?

Plan. Compare the sector angle with a full turn and simplify.

fraction=120360sector angle over full turn=13divide top and bottom by 12013=2×16twice the previous fraction\begin{aligned} \text{fraction}&=\frac{120}{360} && \text{sector angle over full turn}\\ &=\frac13 && \text{divide top and bottom by }120\\ \frac13&=2\times\frac16 && \text{twice the previous fraction} \end{aligned}

Answer. 13\frac13 of the circle.

Check. Three 120∘120^\circ sectors make 360∘360^\circ, so three one-third shares fill the circle.

Counting equal sectors and stating one sector's share are reverse questions.

Watch out

360120=3\frac{360}{120}=3 counts sectors; it is not the fraction occupied by one sector. One of three equal sectors occupies 13\frac13.

Find the share before calculating any lengths or areas.

Arc lengths and sector perimeters

An arc is the sector's share of the whole circumference. A perimeter is the complete boundary of the sector region. After following its arc, you must travel along both radii to return to your starting point.

You already know

A circle of radius rr has circumference 2πr2\pi r. Multiply by a fraction to find that share of the whole. Keep π\pi when the answer must be exact.

For example, one sixth of a circumference 12π cm12\pi\,\mathrm{cm} is 2π cm2\pi\,\mathrm{cm}. The cut edges have not become shorter fractions of a radius: each is still a whole radius.

Arc length and sector perimeter

For a sector of positive radius rr and angle 0∘<θ<360∘0^\circ<\theta<360^\circ, let f=θ/360∘f=\theta/360^\circ. Then

L=f(2πr),P=L+2r.L=f(2\pi r),\qquad P=L+2r.

LL is arc length; PP is the sector's total perimeter. Here the angle divides 360∘360^\circ into equal whole-number shares.

These sector formulas are not given in the List of formulas; the full-circle circumference formula is given.

The angle condition matters. A full 360∘360^\circ circle has no two cut edges in its outside boundary, so its perimeter is just the circumference.

[DIAGRAM: asset_name: Arc versus full sector boundary; asset_slug: arc_sector_boundary; caption: Follow the curved arc and then both radii to trace the complete boundary.; description: Two exact sectors of radius 6 centimetres have angles 60 and 120 degrees. Each arc is blue, its two radial edges are grey and labelled 6 centimetres, and a separate angle arc marks the central angle. Labels distinguish curved arc L from total perimeter L plus 12.]
Follow the curved arc and then both radii to trace the complete boundary.

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