Arc length and sector area Problem set A One question at a time

A sector has radius 7.2 cm and angle 115° at the centre. Find the length of its arc.

• \(s=\frac{115}{360}\times2\pi\times7.2=14.45\)
• 14.5 cm

A sector has radius 11 cm and angle 72° at the centre. Find its area.

• \(A=\frac{72}{360}\times\pi\times11^2=76.03\)
• 76.0 cm²

An arc of length 15 cm subtends an angle of 70° at the centre of a circle. Find the radius of the circle.

• \(15=\frac{70}{360}\times2\pi r\)
• \(r=\frac{15\times360}{70\times2\pi}=12.28\)
• r = 12.3 cm

A sector of radius 7 cm has area 40 cm². Find the angle at the centre.

• \(40=\frac{\theta}{360}\times\pi\times7^2\)
• \(\theta=\frac{40\times360}{\pi\times7^2}=93.54\)
• θ = 93.5°

A sector has radius 8 cm and angle 0.7 radians at the centre. Find

1. (a) the arc length

   • \(s=r\theta=8\times0.7=5.60\) cm

2. (b) the area of the sector

   • \(A=\frac{1}{2}r^2\theta=\frac{1}{2}\times8^2\times0.7=22.4\) cm²

A designer cuts a segment from a circular metal disc of radius 9 cm. The straight edge AB subtends an angle of 75° at the centre O. Find

1. (a) the perimeter of the segment

   • \(\text{arc}=\frac{75}{360}\times2\pi\times9=11.78\)
   • \(AB^2=9^2+9^2-2\times9\times9\cos75^\circ,\; AB=10.96\)
   • \(\text{perimeter}=11.78+10.96=22.7\) cm

2. (b) the area of the segment

   • \(\text{sector}=\frac{75}{360}\times\pi\times9^2=53.01\)
   • \(\text{triangle}=\frac{1}{2}\times9^2\sin75^\circ=39.12\)
   • \(\text{segment}=53.01-39.12=13.9\) cm²

A lawn sprinkler sprays water up to 6.5 m and sweeps through 130°. Find

1. (a) the area of the region

   • \(A=\frac{130}{360}\times\pi\times6.5^2=47.93\) m²

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

   • \(s=\frac{130}{360}\times2\pi\times6.5=14.75\)
   • \(\text{edge}=s+2r=14.75+13=27.7\) m

A horse is tied by an 8 m rope to the outside corner of a barn that is 11 m by 14 m. The horse can graze only outside the barn. Find the area it can reach.

• The shed fills a right angle at the corner, so the rope swings through 360° − 90° = 270°.
• \(A=\frac{270}{360}\times\pi\times8^2=\frac{3}{4}\pi\times64=150.8\)
• 151 m² (8 m is shorter than both walls, so the rope never wraps round the shed)

A point P on the rim of a bicycle wheel of radius 0.34 m moves in a circle about the axle. The wheel turns at 9.5 radians per second.

1. (a) Find the distance P travels in 4 seconds.

   • \(\theta=9.5\times4=38\) radians
   • \(s=r\theta=0.34\times38=12.920\), so 12.9 m

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

   • \(\text{speed}=r\omega=0.34\times9.5=3.230\) m s⁻¹
   • \(3.230\times3.6=11.63\) km/h, so 11.6 km/h

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

   • \(\text{time}=\frac{2\pi}{9.5}=0.661\), so 0.661 s

A circular pond has radius 10 m. A straight path AB of length 14 m cuts off a segment of the pond. Find

1. (a) the angle AOB at the centre

   • \(\sin\left(\frac{\theta}{2}\right)=\frac{7}{10}=0.7\)
   • \(\frac{\theta}{2}=44.43^\circ,\; \theta=88.9^\circ\)

2. (b) the area of the segment

   • \(\text{sector}=\frac{88.85}{360}\times\pi\times10^2=77.54\)
   • \(\text{triangle}=\frac{1}{2}\times10^2\times\sin88.85^\circ=49.99\)
   • \(\text{segment}=27.5\) m²

A sector has a perimeter of 20 cm and an angle of 1.2 radians at the centre. Find

1. (a) the radius r

   • \(\text{perimeter}=2r+r\theta=r(2+1.2)=20\)
   • \(r=\frac{20}{3.2}=6.25\) cm

2. (b) the area of the sector

   • \(A=\frac{1}{2}r^2\theta=\frac{1}{2}\times6.2500^2\times1.2=23.4\) cm²