Faded examples · Finding the arc length of a sector

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Each worked example shows a little less than the one before — you complete the faded (blank) steps yourself, following the same four steps every time. In each example also answer the check question — it tests why each step works.

①Example 1fully worked — read it through

Sector with $\theta = 90°$ and $r = 8$ cm.

1Fraction of the circle

$\dfrac{90}{360}=\dfrac{1}{4}$

2Multiply by the circumference $2\pi r$

arc $=\dfrac{1}{4}\times 2\pi r$

3Substitute $r$ and simplify

$=\dfrac{1}{4}\times 2\pi\times 8=\dfrac{1}{4}\times 16\pi$

4Work out the answer

$=4\pi=12.6$ cm (3 s.f.)

Check · Step 2 multiplies the fraction by $2\pi r$. What does $2\pi r$ represent?

A the circumference of the whole circleB the area of the whole circleC the diameter of the circleD the perimeter of the sector

②Example 2you finish the last 1 step

Sector with $\theta = 120°$ and $r = 9$ cm.

1Fraction of the circle

$\dfrac{120}{360}=\dfrac{1}{3}$

2Multiply by the circumference $2\pi r$

arc $=\dfrac{1}{3}\times 2\pi r$

3Substitute $r$ and simplify

$=\dfrac{1}{3}\times 2\pi\times 9=\dfrac{1}{3}\times 18\pi$

4Work out the answer

Check · A student wrote the fraction as $\dfrac{120}{180}$. What did they do wrong?

A a full circle is $360°$, not $180°$B nothing — $\dfrac{120}{180}$ is correctC they should divide by the radiusD they should multiply $120$ by $360$

③Example 3you finish the last 2 steps

Sector with $\theta = 45°$ and $r = 12$ cm.

1Fraction of the circle

$\dfrac{45}{360}=\dfrac{1}{8}$

2Multiply by the circumference $2\pi r$

arc $=\dfrac{1}{8}\times 2\pi r$

3Substitute $r$ and simplify

4Work out the answer

Check · Why do we use $r = 12$ in $2\pi r$, and not the diameter $24$?

A the formula uses the radius, and $12$ is the radiusB the radius is always half of $\theta$C $24$ is too large to useD either works — it makes no difference

④Example 4your turn — every step

Sector with $\theta = 60°$ and $r = 15$ cm.

1Fraction of the circle

2Multiply by the circumference $2\pi r$

3Substitute $r$ and simplify

4Work out the answer

Check · The exact answer is $5\pi$ cm. When is leaving it "in terms of $\pi$" the best form?

A when the question says "in terms of $\pi$"B always — never give a decimalC only for areas, not lengthsD only when $\theta$ is a multiple of $90°$

Answers · Arc Length & Sector Area

Faded examples

① Example 1 $\dfrac{90}{360}=\dfrac{1}{4}$→arc $=\dfrac{1}{4}\times 2\pi r$→$=\dfrac{1}{4}\times 2\pi\times 8=\dfrac{1}{4}\times 16\pi$→$=4\pi=12.6$ cm (3 s.f.) $4\pi = 12.6 cm (3 s.f.)$

Check: A — the circumference of the whole circle

② Example 2 $\dfrac{120}{360}=\dfrac{1}{3}$→arc $=\dfrac{1}{3}\times 2\pi r$→$=\dfrac{1}{3}\times 2\pi\times 9=\dfrac{1}{3}\times 18\pi$→$=6\pi=18.8$ cm (3 s.f.) $6\pi = 18.8 cm (3 s.f.)$

Check: A — a full circle is $360°$, not $180°$

③ Example 3 $\dfrac{45}{360}=\dfrac{1}{8}$→arc $=\dfrac{1}{8}\times 2\pi r$→$=\dfrac{1}{8}\times 2\pi\times 12=\dfrac{1}{8}\times 24\pi$→$=3\pi=9.42$ cm (3 s.f.) $3\pi = 9.42 cm (3 s.f.)$

Check: A — the formula uses the radius, and $12$ is the radius

④ Example 4 $\dfrac{60}{360}=\dfrac{1}{6}$→arc $=\dfrac{1}{6}\times 2\pi r$→$=\dfrac{1}{6}\times 2\pi\times 15=\dfrac{1}{6}\times 30\pi$→$=5\pi=15.7$ cm (3 s.f.) $5\pi = 15.7 cm (3 s.f.)$

Check: A — when the question says "in terms of $\pi$"