Circular measure: แบบฝึกหัด

An arc of a circle of radius r cm subtends an angle of \( 45^\circ \) at the centre. Given that the length of the arc is \( 4\pi \) cm, find the value of r.

A sector of a circle has an arc length of \(10\) cm and a radius of \(5\) cm. Calculate the area of the sector.

A sector of a circle with centre O has radius \(R\) and sector angle \(2\alpha\) radians. A smaller circle with radius \(r\) is inscribed inside the sector so that it is tangent to both radii OA and OB, and also tangent to the arc AB. Which of the following expressions correctly gives \(r\) in terms of \(R\) and \(\alpha\)?

A sector of a circle has a radius of \(5\) cm and an angle of \(1.2\) radians at the centre. Calculate the length of the arc of this sector.

In the diagram, OAB is a sector of a circle with centre O and radius 4 cm. The angle AOB is \(\frac{\pi}{3}\) radians. Calculate the area of the minor segment bounded by the chord AB and the arc AB.

A sector of a circle has a radius of \( 6 \) cm and an arc length of \( 9 \) cm. Find the angle of the sector in radians.

A sector of a circle has an arc length of \( 10 \text{ cm} \) and a radius of \( 4 \text{ cm} \). Calculate the area of this sector in \( \text{cm}^2 \).

A sector of a circle with centre O and radius \(r\) has a sector angle of \(\theta\) radians. Given that the area of the segment created by the chord joining the ends of the arc is equal to one-third of the area of the sector, show that \(2\theta = 3\sin\theta\).

The diagram shows a sector OAB of a circle with centre O and radius \( 9 \text{ cm} \). The arc length AB is \( 12 \text{ cm} \).

(a) Find the angle AOB in radians.
(b) Calculate the perimeter of the sector OAB.

The diagram shows a triangle OPQ where OP = 8 cm, OQ = 10 cm and angle POQ = \( 0.6 \text{ radians} \). A sector OPR is drawn with centre O and radius 8 cm, where R lies on OQ.

(a) Calculate the area of the sector OPR.
(b) Calculate the area of the triangle OPQ.
(c) Find the area of the region bounded by the arc PR and the lines RQ and QP.