Radian measure and arc length

Discover

Wrap a piece of string as long as the radius around the edge of a circle. The angle it covers at the center is the same for every circle. How many of these angles make a half turn?

About 3.14, that is $\pi$
180

Key idea

One radian is the angle that cuts off an arc as long as the radius.

180^\circ = \pi

radians. Degrees to radians: multiply by

\dfrac{\pi}{180}

. Radians to degrees: multiply by

\dfrac{180}{\pi}

. Arc length:

s = r\theta

, with

\theta

in radians.

Worked example

Convert $135^\circ$ to radians, then find the arc it cuts off on a circle of radius 4 cm.

135 \cdot \dfrac{\pi}{180} = \dfrac{3\pi}{4}

Multiply by π/180.

s = 4 \cdot \dfrac{3\pi}{4} = 3\pi

cm

s = rθ, θ in radians.

Your turn, guided

Your turn: convert $\dfrac{5\pi}{6}$ radians to degrees.

\dfrac{5 \cdot 180}{6}

Replace π by 180°.

= 150^\circ

900 ÷ 6.

Practice ★

Convert $225^\circ$ to radians. Give the exact value.
Convert $330^\circ$ to radians. Give the exact value.
Convert $300^\circ$ to radians. Give the exact value.
Convert $30^\circ$ to radians. Give the exact value.
Convert $180^\circ$ to radians. Give the exact value.

Watch out!

Practice ★★

Convert $\pi$ radians to degrees.
Convert $\frac{5 \pi}{4}$ radians to degrees.
Convert $3 \pi$ radians to degrees.

A circle has radius 4 cm. What is the length of the arc cut off by a central angle of $\frac{3 \pi}{4}$ ? Give the exact value, in cm.
A circle has radius 8 cm. What is the length of the arc cut off by a central angle of $\frac{2 \pi}{3}$ ? Give the exact value, in cm.
A circle has radius 12 cm. What is the length of the arc cut off by a central angle of $\frac{5 \pi}{6}$ ? Give the exact value, in cm.

Exit ticket

Three questions to check you have got it.

A circle has radius 4 cm. What is the length of the arc cut off by a central angle of $\frac{3 \pi}{4}$ ? Give the exact value, in cm.
A circle has radius 8 cm. What is the length of the arc cut off by a central angle of $\frac{2 \pi}{3}$ ? Give the exact value, in cm.
A circle has radius 12 cm. What is the length of the arc cut off by a central angle of $\frac{5 \pi}{6}$ ? Give the exact value, in cm.

Summary card

Radians

$180^\circ = \pi$ rad.
Degrees → radians: $\times \dfrac{\pi}{180}$ . Radians → degrees: $\times \dfrac{180}{\pi}$ .
Arc length: $s = r\theta$ , $\theta$ in radians.

Watch out: never put degrees into $s = r\theta$ .

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