1fully worked

A sector with $\theta = 90°$ has arc length $6\pi$ cm. Find $r$.

1$6\pi=\dfrac{90}{360}\times 2\pi r$Write the arc-length equation

2$6\pi=\dfrac{1}{4}\times 2\pi r=\dfrac{\pi r}{2}$Simplify the fraction side

3$r=\dfrac{2\times 6\pi}{\pi}$Rearrange to find $r$

4$r=12$ cmWork out the answer

Check - Why can we set $6\pi$ equal to $\dfrac{90}{360}\times 2\pi r$?

A the arc-length formula still applies: we know the arc but not $r$B because $6\pi$ is the radiusC because all sectors have arc $6\pi$D we cannot: the formula only finds arcs

2you finish it

A sector with $\theta = 120°$ has arc length $10\pi$ cm. Find $r$.

1$10\pi=\dfrac{120}{360}\times 2\pi r$Write the arc-length equation

2$10\pi=\dfrac{1}{3}\times 2\pi r=\dfrac{2\pi r}{3}$Simplify the fraction side

3$r=\dfrac{3\times 10\pi}{2\pi}$Rearrange to find $r$

4Work out the answer

Check - A student divided $10\pi$ by $2\pi$ and got $r=5$. What went wrong?

A they ignored the fraction $\frac{1}{3}$: the arc is only part of the circumferenceB nothing: $r=5$ is correctC they should have multiplied by $2\pi$D they used the area formula

3you finish it

A sector with $\theta = 60°$ has arc length $3\pi$ cm. Find $r$.

1$3\pi=\dfrac{60}{360}\times 2\pi r$Write the arc-length equation

2$3\pi=\dfrac{1}{6}\times 2\pi r=\dfrac{\pi r}{3}$Simplify the fraction side

3Rearrange to find $r$

4Work out the answer

Check - In step 3 the $\pi$ cancels. Why is that allowed?

A both sides are divided by $\pi$, so it cancels exactlyB $\pi$ is approximately 3, so it rounds awayC it is not allowed: the answer keeps $\pi$D only the left side is divided by $\pi$

4you do it all

A sector with $\theta = 45°$ has arc length $2\pi$ cm. Find $r$.

1Write the arc-length equation

2Simplify the fraction side

3Rearrange to find $r$

4Work out the answer

Check - How could you CHECK the answer $r=8$?

A substitute it back: $\frac{1}{8}\times 2\pi\times 8=2\pi$ ✓B square it and compare with the areaC double it and compare with $\theta$D no check is possible without a calculator