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AI SL · 12.10

Geometry & Trigonometry

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12.10.10

Arc length and sector area

12.10.10

Arc length and sector area

Target: I can find the length of an arc and the area of a sector when the angle at the centre is in degrees. (+4 more)

By the end of this section

Key points

Arcs and sectors, angle in degrees

  • A sector is the fraction \(\frac{\theta}{360}\) of a whole circle. Take the same fraction of the circumference or of the area.
  • Arc length: \(s=\frac{\theta}{360}\times 2\pi r\). Sector area: \(A=\frac{\theta}{360}\times\pi r^2\).
  • The angle is the angle at the centre. The radius is the distance from the centre, not the diameter.
  • The perimeter of a sector is the arc plus two radii: \(s+2r\).
  • Arc length is a length, so cm. Area is cm². Do not mix the two formulas: length has \(2\pi r\), area has \(\pi r^2\).

Segments

  • A segment is the region between a chord and its arc.
  • Perimeter of a segment = arc + chord. There are no radii in it.
  • Area of a segment = area of the sector − area of the triangle. The triangle has two sides r and the angle θ between them: area \(=\frac{1}{2}r^2\sin\theta\).
  • Find the chord with the cosine rule: \(AB^2=r^2+r^2-2r^2\cos\theta\).

A tethered animal at a corner

  • A rope fixed at the corner of a shed swings round the outside of the shed. The angle outside a right-angled corner is 360° − 90° = 270°.
  • That is three-quarters of a circle: area \(\frac{270}{360}\pi r^2=\frac{3}{4}\pi r^2\). This works while the rope is no longer than either wall.
  • Sketch the corner, the rope and the region first.

Explore: arc length and sector area

Set the radius with the slider and drag the point to set the angle at the centre. The arc length and the area are worked out with the numbers.



Fraction of the circle
Arc length, s
Sector area, A
Perimeter of the sector

Drag the point to change the angle.

  1. Double the radius and keep the angle. What happens to the arc length? What happens to the area? Why are they different?
  2. Set the angle to 180°. What fraction of the circle is the sector? Check the area against half of πr².
  3. Tick the segment box. Why is the segment area the sector area minus the triangle area?
  4. Find an angle and a radius that give a sector with an area of exactly 10π cm². Is there more than one pair?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Arc length from degrees

A sector of a circle has radius 6.5 cm and angle 65° at the centre. Find the length of the arc.

65°6.5 cmOAB
Working
  1. The fraction

    \(\frac{65}{360}\)

    The sector is this fraction of the whole circle.

  2. Arc length

    \(s=\frac{65}{360}\times2\pi\times6.5=7.374\)

    Fraction of the circumference \(2\pi r\).

  3. Answer

    7.37 cm

    3 significant figures, with the unit.

Video walkthrough coming
Area of a sector

A sector has radius 9 cm and angle 140° at the centre. Find its area.

140°9 cmOAB
Working
  1. The fraction

    \(\frac{140}{360}\)

    The same fraction, now of the circle's area.

  2. Sector area

    \(A=\frac{140}{360}\times\pi\times9^2=98.96\)

    Fraction of \(\pi r^2\).

  3. Answer

    99.0 cm²

    Area, so cm².

Video walkthrough coming
Perimeter of a segment

The diagram shows a sector of a circle with centre O, radius 12 cm and angle AOB = 50°. The chord AB cuts off a segment. Find the perimeter of the segment.

50°12 cmOAB
Working
  1. Arc

    \(\frac{50}{360}\times2\pi\times12=10.47\)

    The curved edge.

  2. Chord AB

    \(AB^2=12^2+12^2-2\times12\times12\cos50^\circ,\; AB=10.14\)

    Cosine rule in triangle OAB. Calculator in degree mode.

  3. Add

    \(10.47+10.14=20.6\) cm

    The straight edge and the curved edge.

Video walkthrough coming
Area of a segment

For the same segment (radius 12 cm, angle AOB = 50°), find the area of the segment.

50°12 cmOAB
Working
  1. Sector

    \(\frac{50}{360}\times\pi\times12^2=62.83\)

    Area of sector OAB.

  2. Triangle

    \(\frac{1}{2}\times12\times12\times\sin50^\circ=55.16\)

    Two sides and the angle between them.

  3. Subtract

    \(62.83-55.16=7.68\) cm²

    Segment = sector − triangle.

Video walkthrough coming
A tethered animal

A goat is tied by a 7 m rope to the corner X of a shed that is 9 m by 12 m. The goat can graze only outside the shed. Find the area it can reach.

270°7 mX
Working
  1. The angle

    \(360^\circ-90^\circ=270^\circ\)

    The shed fills a right angle at X, so the rope swings through the rest.

  2. The area

    \(A=\frac{270}{360}\times\pi\times7^2=\frac{3}{4}\pi\times49=115.5\)

    The rope (7 m) is shorter than both walls (9 m and 12 m), so it never wraps round the shed.

  3. Answer

    115 m²

    Area, so m².

Video walkthrough coming