The angle at the centre subtended by an arc equal in length to the radius.

60^\circ, 45^\circ, 30^\circ in radians?

\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6}

Why must \theta be in radians for s=r\theta?

The formula comes from the definition of a radian; with degrees an extra factor of \frac{\pi}{180} would be needed.

Radian measure, arc length and sector areaAQA A-Level Maths: Flashcards

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How many radians in $360^\circ$?

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How many radians in $360^\circ$ ?: $2\pi$ radians
Convert degrees to radians.: Multiply by $\frac{\pi}{180}$
Convert radians to degrees.: Multiply by $\frac{180}{\pi}$
Define one radian.: The angle at the centre subtended by an arc equal in length to the radius.
Arc length of a sector, $\theta$ in radians?: $s=r\theta$
Area of a sector, $\theta$ in radians?: $A=\frac12r^2\theta$
Area of a sector in terms of $r$ and $s$ ?: $A=\frac12rs$
Perimeter of a sector?: $2r+r\theta$
Area of a minor segment?: $\frac12r^2(\theta-\sin\theta)$
Length of a chord subtending $\theta$ at the centre?: $2r\sin\frac{\theta}{2}$
$60^\circ$ , $45^\circ$ , $30^\circ$ in radians?: $\frac{\pi}{3}$ , $\frac{\pi}{4}$ , $\frac{\pi}{6}$
Area of a ring-shaped sector between radii $R$ and $r$ ?: $\frac12\theta(R^2-r^2)$
Why must $\theta$ be in radians for $s=r\theta$ ?: The formula comes from the definition of a radian; with degrees an extra factor of $\frac{\pi}{180}$ would be needed.

Exam questions on Radian measure, arc length and sector area

A sector $OAB$ has centre $O$ , radius $5$ cm and $A\hat{O}B=1.2$ radians.
Convert $1.2$ radians to degrees, giving your answer to 1 decimal place.2 marks
A sector of a circle has arc length $8$ cm and area $24$ cm $^2$ .
Find the perimeter of the sector.2 marks
A circle has centre $O$ and radius $10$ cm. The chord $AB$ subtends an angle of $\frac{\pi}{3}$ radians at $O$ . The minor segment is the region between the chord $AB$ and the minor arc $AB$ .
Find the exact perimeter of the minor segment.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).