Radians

Convert between degrees and radians, apply radian measure to find arc length and sector area, and calculate segment area in composite problems.

Worked examples

Converting between degrees and radians

Straightforward

Problem

Convert (a)

150°

to radians and (b)

\dfrac{7\pi}{6}

radians to degrees.

Finding arc length and sector area

Moderate

Problem

A sector has radius

9

cm and central angle

\dfrac{5\pi}{6}

radians. Find (a) the arc length and (b) the area of the sector. Give answers to 2 decimal places.

Segment area and arc length — composite problem

Challenging

Problem

A circle has radius

7

cm. A chord subtends a central angle of

\dfrac{\pi}{2}

radians. Find (a) the area of the minor segment and (b) the arc length of the minor arc. Give answers to 2 decimal places.

Practise

Q1·Straightforward

Convert

135°

to radians.

$\dfrac{\pi}{4}$$\dfrac{3\pi}{4}$$\dfrac{\pi}{2}$$\dfrac{2\pi}{3}$

Q2·Straightforward

Convert

\dfrac{5\pi}{6}

radians to degrees.

Q3·Straightforward

Convert

240°

to radians.

$\dfrac{4\pi}{3}$$\dfrac{5\pi}{6}$$\dfrac{2\pi}{3}$$\dfrac{3\pi}{4}$

Q4·Straightforward

Convert

\dfrac{\pi}{4}

radians to degrees.

Q5·Moderate

A circle has radius

8

cm and a sector with central angle

\dfrac{\pi}{4}

radians. Find the arc length of the sector, to 2 decimal places.

Q6·Moderate

A sector has radius

6

cm and central angle

\dfrac{2\pi}{3}

radians. Find the area of the sector, to 2 decimal places.

Q7·Moderate

A circle of radius

12

cm has a sector with arc length

9\pi

cm. Find the central angle of the sector in radians, to 2 decimal places.

Q8·Moderate

A sector has radius

10

m and central angle

\dfrac{3\pi}{5}

radians. Find its arc length, to 2 decimal places.

Q9·Challenging

A sector has radius

5

cm and central angle

\dfrac{\pi}{3}

radians. Find the area of the minor segment (the region between the chord and the arc), to 2 decimal places.

Q10·Challenging

A circle has radius

9

cm. A chord subtends a central angle of

\dfrac{2\pi}{3}

radians. Find the area of the minor segment, to 2 decimal places.

Q11·Challenging

A sector has radius

12

cm and arc length

8\pi

cm. Find the area of the sector, to 2 decimal places.

Q12·Challenging

A sector has radius

6

cm and area

15\pi

cm². Find the arc length of the sector, to 2 decimal places.