Circular measure：練習題

A sector of a circle with radius \(r\) cm has an area of \(50\text{ cm}^2\). Given that the perimeter of the sector is \(30\text{ cm}\), find the two possible values of \(r\).

In a circle of radius \(r\), a chord \(AB\) has length \(r\). Find 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 area of the sector.

In the triangle \(OAB\), \(\angle OAB = \frac{\pi}{2}\) and \(\angle AOB = \alpha\) radians. A circular arc \(AC\), with centre \(O\) and radius \(OA = r\), is drawn such that \(C\) lies on the line \(OB\). Find the area of the region bounded by the arc \(AC\) and the line segments \(AB\) and \(BC\).

A sector of a circle with radius \(r\) cm has a perimeter of \(4r\) cm. Find the area of the sector in terms of \(r\).

An arc of a circle subtends an angle of \( \frac{2\pi}{5} \) radians at the centre. If the arc length is 10 cm, find the radius of the circle.

A uniform wire of length 30 cm is bent to form a sector of a circle. Show that the area of the sector is \(A = 15r - r^2\), and find the maximum possible area.

A sector of a circle has radius 8 cm and area 48 cm². Find the angle of the sector in radians.

The diagram shows a sector OAB of a circle with centre O and radius r cm. The angle AOB is \(\theta\) radians. The perimeter of the sector is 20 cm and the area of the sector is 24 cm \(^2\) .

(a) Show that r satisfies the equation \(r^2 - 20r + 48 = 0\) .

(b) Find the possible values of r and the corresponding values of \(\theta\) .

The diagram shows a sector OAB of a circle with centre O and radius r cm. The angle AOB is \(\theta\) radians. The point C lies on OB such that \(OC = \frac{1}{2}r\) . The line AC is a straight line.

(a) Given that the area of the sector OAB is 20 cm \(^2\) and the area of the triangle OAC is 8 cm \(^2\) , find the values of r and \(\theta\) .

(b) Find the perimeter of the shaded region ABC, which is bounded by the arc AB, the line segment BC, and the line segment CA.