Faded examples · Finding the arc length of a sector

Calculator

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

$=6\pi=18.8$ cm (3 s.f.)

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

$=\dfrac{1}{8}\times 2\pi\times 12=\dfrac{1}{8}\times 24\pi$

4Work out the answer

$=3\pi=9.42$ cm (3 s.f.)

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

$\dfrac{60}{360}=\dfrac{1}{6}$

2Multiply by the circumference $2\pi r$

arc $=\dfrac{1}{6}\times 2\pi r$

3Substitute $r$ and simplify

$=\dfrac{1}{6}\times 2\pi\times 15=\dfrac{1}{6}\times 30\pi$

4Work out the answer

$=5\pi=15.7$ cm (3 s.f.)

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 · Finding the arc length of a sector

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 $\pi$"

Faded examples · Finding the area of a sector

Calculator

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 $r = 6$ cm.

1Fraction of the circle

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

2Multiply by the area $\pi r^2$

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

3Substitute $r$ and simplify

$=\dfrac{1}{4}\times \pi\times 6^2=\dfrac{1}{4}\times 36\pi$

4Work out the answer

$=9\pi=28.3$ cm² (3 s.f.)

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

A the area of the whole circleB the circumference of the whole circleC the area of the sectorD the perimeter of the sector

②Example 2you finish the last 1 step

Sector with $\theta = 120°$ and $r = 6$ cm.

1Fraction of the circle

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

2Multiply by the area $\pi r^2$

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

3Substitute $r$ and simplify

$=\dfrac{1}{3}\times \pi\times 6^2=\dfrac{1}{3}\times 36\pi$

4Work out the answer

$=12\pi=37.7$ cm² (3 s.f.)

Check · A student wrote area $=\dfrac{1}{3}\times 2\pi r$. What have they actually found?

A the arc length, not the areaB the same thing: both formulas workC the area of the whole circleD the diameter of the circle

③Example 3you finish the last 2 steps

Sector with $\theta = 45°$ and $r = 8$ cm.

1Fraction of the circle

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

2Multiply by the area $\pi r^2$

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

3Substitute $r$ and simplify

$=\dfrac{1}{8}\times \pi\times 8^2=\dfrac{1}{8}\times 64\pi$

4Work out the answer

$=8\pi=25.1$ cm² (3 s.f.)

Check · In step 3 we work out $8^2=64$. Why is the radius squared?

A the area formula is $\pi r^2$, so $r$ is squaredB to make the answer biggerC because a sector has two radiiD it should be doubled, not squared

④Example 4your turn: every step

Sector with $\theta = 60°$ and $r = 9$ cm.

1Fraction of the circle

$\dfrac{60}{360}=\dfrac{1}{6}$

2Multiply by the area $\pi r^2$

area $=\dfrac{1}{6}\times \pi r^2$

3Substitute $r$ and simplify

$=\dfrac{1}{6}\times \pi\times 9^2=\dfrac{1}{6}\times 81\pi$

4Work out the answer

$=13.5\pi=42.4$ cm² (3 s.f.)

Check · The units of the answer are cm². Why squared units?

A area is measured in square unitsB because the radius was squaredC all sector answers use cm²D the angle is in square degrees

Answers · Arc Length & Sector Area

Faded examples · Finding the area of a sector

① Example 1 $\dfrac{90}{360}=\dfrac{1}{4}$→area $=\dfrac{1}{4}\times \pi r^2$→$=\dfrac{1}{4}\times \pi\times 6^2=\dfrac{1}{4}\times 36\pi$→$=9\pi=28.3$ cm² (3 s.f.) $9\pi = 28.3 cm² (3 s.f.)$

Check: A: the area of the whole circle

② Example 2 $\dfrac{120}{360}=\dfrac{1}{3}$→area $=\dfrac{1}{3}\times \pi r^2$→$=\dfrac{1}{3}\times \pi\times 6^2=\dfrac{1}{3}\times 36\pi$→$=12\pi=37.7$ cm² (3 s.f.) $12\pi = 37.7 cm² (3 s.f.)$

Check: A: the arc length, not the area

③ Example 3 $\dfrac{45}{360}=\dfrac{1}{8}$→area $=\dfrac{1}{8}\times \pi r^2$→$=\dfrac{1}{8}\times \pi\times 8^2=\dfrac{1}{8}\times 64\pi$→$=8\pi=25.1$ cm² (3 s.f.) $8\pi = 25.1 cm² (3 s.f.)$

Check: A: the area formula is $\pi r^2$, so $r$ is squared

④ Example 4 $\dfrac{60}{360}=\dfrac{1}{6}$→area $=\dfrac{1}{6}\times \pi r^2$→$=\dfrac{1}{6}\times \pi\times 9^2=\dfrac{1}{6}\times 81\pi$→$=13.5\pi=42.4$ cm² (3 s.f.) $13.5\pi = 42.4 cm² (3 s.f.)$

Check: A: area is measured in square units

Faded examples · Finding the perimeter of a sector

Calculator

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 $r = 10$ cm.

1Fraction of the circle

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

2Find the arc length

arc $=\dfrac{1}{4}\times 2\pi\times 10=5\pi$

3Add the two radii

$P=5\pi+10+10=5\pi+20$

4Work out the answer

$=35.7$ cm (3 s.f.)

Check · Step 3 adds $20$. Where does the $20$ come from?

A the two straight radius edges, $10+10$B the diameter of the whole circleC double the angleD the arc length rounded up

②Example 2you finish the last 1 step

Sector with $\theta = 120°$ and $r = 6$ cm.

1Fraction of the circle

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

2Find the arc length

arc $=\dfrac{1}{3}\times 2\pi\times 6=4\pi$

3Add the two radii

$P=4\pi+6+6=4\pi+12$

4Work out the answer

$=24.6$ cm (3 s.f.)

Check · A student’s answer was just $4\pi \approx 12.6$ cm. What did they forget?

A the two straight edges: they found only the arcB nothing: that is the perimeterC to double the arc lengthD to halve the arc length

③Example 3you finish the last 2 steps

Sector with $\theta = 45°$ and $r = 8$ cm.

1Fraction of the circle

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

2Find the arc length

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

3Add the two radii

$P=2\pi+8+8=2\pi+16$

4Work out the answer

$=22.3$ cm (3 s.f.)

Check · Why does the perimeter use arc $+\,2r$, not arc $+\,r$?

A a sector has TWO straight edges, and both are radiiB one of the edges is a diameterC $2r$ is the circumferenceD to match the two in $2\pi r$

④Example 4your turn: every step

Sector with $\theta = 60°$ and $r = 12$ cm.

1Fraction of the circle

$\dfrac{60}{360}=\dfrac{1}{6}$

2Find the arc length

arc $=\dfrac{1}{6}\times 2\pi\times 12=4\pi$

3Add the two radii

$P=4\pi+12+12=4\pi+24$

4Work out the answer

$=36.6$ cm (3 s.f.)

Check · The perimeter is bigger than the arc length alone. Why?

A it also includes the two straight edgesB it never is: this is a rounding errorC because the angle is smallD perimeters are always doubled

Answers · Arc Length & Sector Area

Faded examples · Finding the perimeter of a sector

① Example 1 $\dfrac{90}{360}=\dfrac{1}{4}$→arc $=\dfrac{1}{4}\times 2\pi\times 10=5\pi$→$P=5\pi+10+10=5\pi+20$→$=35.7$ cm (3 s.f.) $5\pi + 20 = 35.7 cm (3 s.f.)$

Check: A: the two straight radius edges, $10+10$

② Example 2 $\dfrac{120}{360}=\dfrac{1}{3}$→arc $=\dfrac{1}{3}\times 2\pi\times 6=4\pi$→$P=4\pi+6+6=4\pi+12$→$=24.6$ cm (3 s.f.) $4\pi + 12 = 24.6 cm (3 s.f.)$

Check: A: the two straight edges: they found only the arc

③ Example 3 $\dfrac{45}{360}=\dfrac{1}{8}$→arc $=\dfrac{1}{8}\times 2\pi\times 8=2\pi$→$P=2\pi+8+8=2\pi+16$→$=22.3$ cm (3 s.f.) $2\pi + 16 = 22.3 cm (3 s.f.)$

Check: A: a sector has TWO straight edges, and both are radii

④ Example 4 $\dfrac{60}{360}=\dfrac{1}{6}$→arc $=\dfrac{1}{6}\times 2\pi\times 12=4\pi$→$P=4\pi+12+12=4\pi+24$→$=36.6$ cm (3 s.f.) $4\pi + 24 = 36.6 cm (3 s.f.)$

Check: A: it also includes the two straight edges

Faded examples · Finding the radius from the arc length

Calculator

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

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

1Write the arc-length equation

$6\pi=\dfrac{90}{360}\times 2\pi r$

2Simplify the fraction side

$6\pi=\dfrac{1}{4}\times 2\pi r=\dfrac{\pi r}{2}$

3Rearrange to find $r$

$r=\dfrac{2\times 6\pi}{\pi}$

4Work out the answer

$r=12$ cm

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

②Example 2you finish the last 1 step

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

1Write the arc-length equation

$10\pi=\dfrac{120}{360}\times 2\pi r$

2Simplify the fraction side

$10\pi=\dfrac{1}{3}\times 2\pi r=\dfrac{2\pi r}{3}$

3Rearrange to find $r$

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

4Work out the answer

$r=15$ cm

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

③Example 3you finish the last 2 steps

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

1Write the arc-length equation

$3\pi=\dfrac{60}{360}\times 2\pi r$

2Simplify the fraction side

$3\pi=\dfrac{1}{6}\times 2\pi r=\dfrac{\pi r}{3}$

3Rearrange to find $r$

$r=\dfrac{3\times 3\pi}{\pi}$

4Work out the answer

$r=9$ cm

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 $\pi$

④Example 4your turn: every step

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

1Write the arc-length equation

$2\pi=\dfrac{45}{360}\times 2\pi r$

2Simplify the fraction side

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

3Rearrange to find $r$

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

4Work out the answer

$r=8$ cm

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

Answers · Arc Length & Sector Area

Faded examples · Finding the radius from the arc length

① Example 1 $6\pi=\dfrac{90}{360}\times 2\pi r$→$6\pi=\dfrac{1}{4}\times 2\pi r=\dfrac{\pi r}{2}$→$r=\dfrac{2\times 6\pi}{\pi}$→$r=12$ cm $r = 12 cm$

Check: A: the arc-length formula still applies: we know the arc but not $r$

② Example 2 $10\pi=\dfrac{120}{360}\times 2\pi r$→$10\pi=\dfrac{1}{3}\times 2\pi r=\dfrac{2\pi r}{3}$→$r=\dfrac{3\times 10\pi}{2\pi}$→$r=15$ cm $r = 15 cm$

Check: A: they ignored the fraction $\frac{1}{3}$: the arc is only part of the circumference

③ Example 3 $3\pi=\dfrac{60}{360}\times 2\pi r$→$3\pi=\dfrac{1}{6}\times 2\pi r=\dfrac{\pi r}{3}$→$r=\dfrac{3\times 3\pi}{\pi}$→$r=9$ cm $r = 9 cm$

Check: A: both sides are divided by $\pi$, so it cancels exactly

④ Example 4 $2\pi=\dfrac{45}{360}\times 2\pi r$→$2\pi=\dfrac{1}{8}\times 2\pi r=\dfrac{\pi r}{4}$→$r=\dfrac{4\times 2\pi}{\pi}$→$r=8$ cm $r = 8 cm$

Check: A: substitute it back: $\frac{1}{8}\times 2\pi\times 8=2\pi$ ✓