Cambridge International A Level · Mathematics (9709)

Circular measure: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Circular measure.

10 questions27 marksFree, no account

Question 1

1 mark

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 \(r\).

Question 2

1 mark

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 \(AB\).

Question 3

1 mark

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

Question 4

1 mark

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 \(BC\).

Question 5

1 mark

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

Question 6

3 marks

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.

Write your answer out first, then check it against the worked solution.

Question 7

5 marks

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.

Write your answer out first, then check it against the worked solution.

Question 8

3 marks

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

Write your answer out first, then check it against the worked solution.

Question 9

4 marks

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 \(\theta\) .

Write your answer out first, then check it against the worked solution.

Question 10

7 marks

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.

Write your answer out first, then check it against the worked solution.

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

You've seen the model answer. Now get yours marked.

This page can show you how a good answer looks. It cannot tell you what your answer was missing. thinka marks your written work against the real mark scheme in about 15 seconds.

Get My Answers Marked Back to the notes

Want more questions like these? Get a fresh set on this topic, marked as you go.