Circles (radians, arc length, sector area): notes and practice questions

A garden sprinkler rotates through an angle of 2.5 radians, watering a sector of a circular lawn. The maximum reach of the water from the sprinkler is 15 metres.

(a) Calculate the length of the arc along the edge of the watered lawn.

(b) Determine the total length of the boundary of the watered sector.

(c) Find the area of the garden watered by the sprinkler.

A lighthouse, L, is situated on a cliff. It emits a beam of light that scans a sector of the sea. The diagram below is a top-down view of the area scanned by the light.

The points M and N represent the furthest points on the edge of the scanned area. The angle of the sector, MÊN, is 1.5 radians.

The perimeter of the sector of the sea scanned by the light is 70 km.

(a) Find the maximum distance the light beam reaches from the lighthouse.

(b) Find the length of the arc MN.

(c) Find the area of the sea scanned by the light beam.

Rewrite each of the following angles from radians to degrees.

(a) $\frac{\pi}{4}$

(b) $\frac{2\pi}{3}$

(c) $\frac{5\pi}{6}$

(d) $\frac{7\pi}{4}$

(e) $-\frac{\pi}{9}$

(f) $-\frac{4\pi}{3}$

(g) $-\frac{3\pi}{2}$

(h) $\frac{7\pi}{5}$

A company is designing a new logo in the shape of a sector of a circle with centre O and radius $r$ cm. The angle of the sector is $\theta$ radians.

The perimeter of the logo is 24 cm and its area is 32 cm$^2$.

(a) Show that $r^2 - 12r + 32 = 0$.

(b) Find the two possible values for $r$ and the corresponding values for $\theta$.

The following diagram shows a sector of a circle with centre O and radius r. The points A and B are on the circumference.

The angle $\text{A}\hat{\text{O}}\text{B} = \frac{\pi}{3}$ radians. The area of the shaded sector is $6\pi$.

(a) Find the value of r.

(b) Hence, find the exact perimeter of the non-shaded region.

A decorative paper fan is made in the shape of a sector of a circle with centre O and radius $r$. The angle of the sector is $\theta$ radians.

The arc length of the fan is 24 cm and its area is 180 cm².

(a) Find the value of $r$.

(b) Find the value of $\theta$.

A circular pizza has a radius of $R$ cm. A chef makes a straight cut from point A on the crust to point B on the crust, such that the angle subtended at the centre O, $\text{A}\hat{\text{O}}\text{B}$, is $\theta$ radians, where $0 < \theta < \pi$. This cut creates a circular segment of pizza (the region bounded by the chord AB and the arc AB).

Another cut is made from the centre O to a point C on the crust, such that A, O, and C are collinear (i.e., AC is a diameter). This creates a triangular slice $\text{OBC}$.

Given that the area of the circular segment (bounded by chord AB and arc AB) is equal to the area of the triangular slice $\text{OBC}$, show that $\theta = 2 \sin \theta$.

Hence determine the value of $\theta$.

A craftsman is designing a decorative metal panel. A section of the design involves a circular arc within a larger piece. The arc is part of a circle with centre O and radius $R = 3$ cm. Two points, P and Q, lie on the circle such that the angle POQ is $\theta$ radians, where $0 < \theta < \pi$.

Find the area of the circular segment defined by the arc PQ and the chord PQ, in terms of $\theta$.

The craftsman cuts this segment from a rectangular metal sheet measuring $6$ cm by $5$ cm. The remaining area of the rectangular sheet is $21.3$ cm$^2$.

Find the value of $\theta$.

A rotating radar dish has a radius of $15$ m. The radar beam sweeps through an angle of $1.2$ radians.

(a) Calculate the area of the sector swept by the radar beam.

(b) An obstruction blocks the signal in a triangular region within the sector. This triangular region has vertices at the centre of the dish and the two points on the edge of the dish that define the sweep angle. Calculate the area of this triangular region.

(c) Hence, determine the effective area covered by the radar.