Arc length and sector area Problem set B Worked solutions

A cake slice is a sector of radius 9.4 cm and angle 48° at the centre. Find the length of its curved edge.

• \(s=\frac{48}{360}\times2\pi\times9.4=7.875\)
• 7.87 cm

A fan is a sector with radius 8.5 cm and angle 130° at the centre. Find its area.

• \(A=\frac{130}{360}\times\pi\times8.5^2=81.97\)
• 82.0 cm²

A curved fence of length 22 m is part of a circle and subtends an angle of 105° at the centre. Find the radius of the circle.

• \(22=\frac{105}{360}\times2\pi r\)
• \(r=\frac{22\times360}{105\times2\pi}=12.00\)
• r = 12.0 m

A pizza slice of radius 8 cm has area 55 cm². Find the angle at the centre.

• \(55=\frac{\theta}{360}\times\pi\times8^2\)
• \(\theta=\frac{55\times360}{\pi\times8^2}=98.48\)
• θ = 98.5°

A wedge of a clock face has radius 6 cm and angle 1.4 radians at the centre. Find

1. (a) the arc length

   • \(s=r\theta=6\times1.4=8.40\) cm

2. (b) the area of the sector

   • \(A=\frac{1}{2}r^2\theta=\frac{1}{2}\times6^2\times1.4=25.2\) cm²

A baker cuts a segment from a circular pastry of radius 14 cm. The straight cut AB subtends an angle of 80° at the centre O. Find

1. (a) the perimeter of the segment

   • \(\text{arc}=\frac{80}{360}\times2\pi\times14=19.55\)
   • \(AB^2=14^2+14^2-2\times14\times14\cos80^\circ,\; AB=18.00\)
   • \(\text{perimeter}=19.55+18.00=37.5\) cm

2. (b) the area of the segment

   • \(\text{sector}=\frac{80}{360}\times\pi\times14^2=136.8\)
   • \(\text{triangle}=\frac{1}{2}\times14^2\sin80^\circ=96.51\)
   • \(\text{segment}=136.8-96.51=40.3\) cm²

A lighthouse beam reaches 12 km and sweeps through 110°. Find

1. (a) the area of the region

   • \(A=\frac{110}{360}\times\pi\times12^2=138.2\) km²

2. (b) the total length of its edge (the arc and the two straight sides)

   • \(s=\frac{110}{360}\times2\pi\times12=23.04\)
   • \(\text{edge}=s+2r=23.04+24=47.0\) km

A llama is tied by a 6 m rope to the outside corner of a hut that is 9 m by 8 m. The llama can graze only outside the hut. Find the area it can reach.

• The shed fills a right angle at the corner, so the rope swings through 360° − 90° = 270°.
• \(A=\frac{270}{360}\times\pi\times6^2=\frac{3}{4}\pi\times36=84.82\)
• 84.8 m² (6 m is shorter than both walls, so the rope never wraps round the shed)

The tip P of a ceiling fan blade is 0.22 m from the centre of the fan. The fan turns at 14 radians per second.

1. (a) Find the distance P travels in 3 seconds.

   • \(\theta=14\times3=42\) radians
   • \(s=r\theta=0.22\times42=9.240\), so 9.24 m

2. (b) Find the speed of P in m s⁻¹ and in km/h.

   • \(\text{speed}=r\omega=0.22\times14=3.080\) m s⁻¹
   • \(3.080\times3.6=11.09\) km/h, so 11.1 km/h

3. (c) Find the time for one complete turn.

   • \(\text{time}=\frac{2\pi}{14}=0.449\), so 0.449 s

A circular flower bed has radius 12 m. A straight fence AB of length 16 m cuts off a segment of the bed. Find

1. (a) the angle AOB at the centre

   • \(\sin\left(\frac{\theta}{2}\right)=\frac{8}{12}=0.6667\)
   • \(\frac{\theta}{2}=41.81^\circ,\; \theta=83.6^\circ\)

2. (b) the area of the segment

   • \(\text{sector}=\frac{83.62}{360}\times\pi\times12^2=105.1\)
   • \(\text{triangle}=\frac{1}{2}\times12^2\times\sin83.62^\circ=71.55\)
   • \(\text{segment}=33.5\) m²

A sector has a perimeter of 18 cm and an angle of 0.8 radians at the centre. Find

1. (a) the radius r

   • \(\text{perimeter}=2r+r\theta=r(2+0.8)=18\)
   • \(r=\frac{18}{2.8}=6.43\) cm

2. (b) the area of the sector

   • \(A=\frac{1}{2}r^2\theta=\frac{1}{2}\times6.4286^2\times0.8=16.5\) cm²