Maths · 6 Mensuration

6.3 Circles, arcs and sectors

Before you start

The radius of a circle runs from the centre to the edge, and the diameter is twice the radius.
Angles at a point add up to $360 °$ .
Simplifying a fraction, e.g. $\frac{80}{360} = \frac{2}{9}$ .
Rounding to 3 significant figures, and to 1 decimal place.
Rearranging a simple formula, e.g. making $r$ the subject of $C = 2 π r$ .
Square roots, and the difference between cm (a length) and cm² (an area).

Syllabus objectives covered (2)

Carry out calculations involving the circumference and area of a circle.
Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle.

Part 1 of 7

π and the circumference

Three lengths describe a circle. The radius $r$ goes from the centre to the edge. The diameter $d$ goes straight across through the centre, so $d = 2 r$ . The circumference $C$ is the distance all the way round the edge.
graph diagram — not drawn yet
A circle with its centre marked O. A radius drawn from O to the edge and labelled r. A diameter drawn across the circle through O and labelled d. The curved edge labelled circumference.
The diameter is two radii end to end. The circumference is the curved edge, not a straight line.
Take any circle and divide its circumference by its diameter. You always get the same number, a little more than $3$ . That number is called $π$ (the Greek letter pi). Its decimal digits never end and never repeat: $π = 3.14159 \dots$
So the circumference is always $π$ times the diameter. Because $d = 2 r$ , you can write the same fact in terms of the radius. These are one formula written two ways:
$C = π d = 2 π r$
The syllabus says the formulas for this subtopic are given. On Papers 2 and 4, the List of formulas on page 2 includes the circumference and the area of a circle. On Papers 5 and 6, any formula you need is printed in the question. So you do not memorise the formulas. What you must do is put the right length in for $r$ .
Take a circle of radius $5$ cm. Then $C = 2 \times π \times 5 = 10 π$ cm. This is an exact answer in terms of π: you leave $π$ as a symbol and simplify the number in front of it. On Paper 2 you have no calculator, so this is the form you give.
On a calculator paper, give a non-exact answer to 3 significant figures unless the question asks for something else: $10 π = 31.415 \dots$ , so $C = 31.4$ cm.
Use the $π$ button. If you must type a value, use $3.142$ . The trap is $3.14$ : it is less accurate and can change the third significant figure. For a radius of $8$ cm, $16 π = 50.265 \dots = 50.3$ cm, but $3.14 \times 16 = 50.24 = 50.2$ cm.
Read whether you are given the radius or the diameter. A circle of diameter $12$ cm has $C = π \times 12 = 12 π$ cm. If you treat the $12$ as a radius, you get $24 π$ cm, which is twice the real circumference.

Part 2 of 7

Area: where πr² comes from

Cut a circle into many thin equal slices, like a cake. Lay them side by side, pointing up and down alternately. The shape you get is very close to a rectangle.
graph diagram — not drawn yet
Left: a circle of radius r cut into 16 equal sectors, half of them shaded. Right: the same 16 sectors laid in a row, alternately pointing up and down, forming a near-rectangle; its height labelled r and its length labelled half the circumference = πr.
The slices make a shape close to a rectangle with height r and length πr. More, thinner slices make it closer still.
The height of that rectangle is one radius, $r$ . Its length is half the curved edge, because half the slices point each way. Half the circumference is $\frac{1}{2} \times 2 π r = π r$ .
The area of a rectangle is length × height, so the area of the circle is:
$A = π r \times r = π r^{2}$
Only the radius is squared. $π r^{2}$ means $π \times r \times r$ . For a radius of $6$ cm, $A = π \times 6^{2} = 36 π$ cm², which is $113.097 \dots = 113$ cm² to 3 significant figures.
The trap is putting the diameter into $π r^{2}$ . A circle of diameter $12$ cm has radius $6$ cm, so its area is $36 π$ cm². Using $12$ gives $144 π$ cm², which is four times too big. Halve the diameter first, every time.
Watch the units. A circumference is a length, so it is in cm. An area is a length times a length, so it is in cm².
Because the radius is squared, doubling the radius does not double the area. $π {(2 r)}^{2} = 4 π r^{2}$ , so the area becomes four times as big. The circumference only doubles, because $2 π (2 r) = 2 \times 2 π r$ .

Part 3 of 7

Working back to the radius

A question can give you the circumference or the area and ask for the radius. Start from the formula and undo it.
From the circumference: $C = 2 π r$ means $r$ was multiplied by $2 π$ . So divide by $2 π$ . If $C = 30$ cm:
$r = \frac{30}{2 π} = 4.774 \dots = 4.77 cm$
From the area: to get $A$ , you square $r$ and then multiply by $π$ . Undo these in reverse order: divide by $π$ first, then take the square root. If $A = 30$ cm²:
$r^{2} = \frac{30}{π} = 9.549 \dots r = \sqrt{9.549 \dots} = 3.090 \dots = 3.09 cm$
Without a calculator, an area given in terms of $π$ lets the $π$ cancel. If $A = 49 π$ cm², then $π r^{2} = 49 π$ , so $r^{2} = 49$ and $r = 7$ cm.
If the radius is used again in a later step, carry its full calculator value forward, not the rounded one. Use the answer key or the memory. Rounding early can move the final answer at 3 significant figures, as the next worked example shows.

Part 4 of 7

An arc length is a fraction of the circumference

An arc is part of the circumference. A sector is the slice of the circle between two radii and the arc that joins them. The angle between the two radii, at the centre, is the angle of the sector. Call it $θ$ .
graph diagram — not drawn yet
A circle with centre O. Two radii OA and OB, each labelled r, with the angle θ marked between them at O. The smaller slice between OA, OB and the arc AB is shaded; the arc AB is drawn thicker and labelled arc length.
The shaded sector is the fraction θ/360 of the whole circle, and its arc is the same fraction of the circumference.
A full turn at the centre is $360 °$ . So a sector with angle $θ$ is the fraction $\frac{θ}{360}$ of the whole circle. Its arc is that same fraction of the circumference:
$arc length = \frac{θ}{360} \times 2 π r$
This is not a new formula to learn. It is the circumference, from the List of formulas, multiplied by the fraction of the circle you have. That is exactly what the objective asks you to do.
Take $θ = 72 °$ and $r = 10$ cm. The fraction is $\frac{72}{360} = \frac{1}{5}$ . The whole circumference is $2 π \times 10 = 20 π$ cm, so the arc is $\frac{1}{5} \times 20 π = 4 π$ cm. On a calculator paper that is $12.566 \dots = 12.6$ cm.
Without a calculator, simplify the fraction before you multiply. $\frac{1}{5}$ of $20 π$ is easy. $\frac{72}{360}$ of $20 π$ looks hard, but it is the same thing.
Check with the cases you already know. $θ = 180 °$ gives $\frac{1}{2}$ , the arc of a semicircle. $θ = 90 °$ gives $\frac{1}{4}$ , the arc of a quarter circle. Your fraction must be less than $1$ unless the angle is a full $360 °$ .

Part 5 of 7

Sector area is the same fraction of the area

The sector is the same slice of the whole disc as its arc is of the edge. So its area is the same fraction $\frac{θ}{360}$ of the whole area:
$sector area = \frac{θ}{360} \times π r^{2}$
For the same sector, $θ = 72 °$ and $r = 10$ cm: the whole area is $π \times 10^{2} = 100 π$ cm², so the sector is $\frac{1}{5} \times 100 π = 20 π$ cm², or $62.8$ cm² to 3 significant figures.
One fraction does both jobs. Find $\frac{θ}{360}$ once. Multiply it by $2 π r$ for the arc, or by $π r^{2}$ for the area. To check that you used the right one, look at the units: an arc length is in cm and a sector area is in cm².
The perimeter of a sector is the whole distance round its edge. That edge is the arc and two straight sides, and each straight side is a radius:
$perimeter of a sector = arc length + 2 r$
For the $72 °$ sector of radius $10$ cm, the perimeter is $4 π + 10 + 10 = 4 π + 20$ cm. On a calculator paper that is $32.566 \dots = 32.6$ cm.
In terms of $π$ , the perimeter has two terms. $4 π + 20$ cannot be simplified to $24 π$ , because $20$ is not a multiple of $π$ . It is like $4 x + 20$ , which is not $24 x$ .

Part 6 of 7

Minor and major sectors

Two radii cut a circle into two sectors. The smaller one is the minor sector, and its angle is less than $180 °$ . The larger one is the major sector, and its angle is more than $180 °$ . In the same way, the two arcs are the minor arc and the major arc.
graph diagram — not drawn yet
A circle with centre O and two radii OA and OB. The angle 120° is marked inside the smaller slice, which is lightly shaded and labelled minor sector. The rest of the circle is shaded darker and labelled major sector, with its reflex angle 240° marked at O.
The two angles at O add up to 360°, so the major sector angle is 360° minus the minor one.
The two angles at the centre are angles at a point, so they add up to $360 °$ . A minor sector of $120 °$ leaves a major sector of $360 ° - 120 ° = 240 °$ .
For a major sector, use its own angle in the fraction. With $r = 6$ cm and the minor angle $120 °$ , the major sector is $\frac{240}{360} = \frac{2}{3}$ of the circle:
$major sector area = \frac{2}{3} \times 36 π = 24 π {cm}^{2}$
Another way is the whole circle minus the minor sector. The minor sector is $\frac{1}{3} \times 36 π = 12 π$ , and $36 π - 12 π = 24 π$ . Both ways agree, which is a good check.
The trap is a diagram that marks only the minor angle. Before you write any fraction, find out which sector the question asks about. If it says major, find $360 ° - θ$ first.
The perimeter of a major sector is the major arc plus the same two radii. With $r = 6$ cm and a $240 °$ angle, the major arc is $\frac{2}{3} \times 12 π = 8 π$ cm, so the perimeter is $8 π + 12$ cm.

Part 7 of 7

Finding the angle or the radius

Sometimes you are given the arc length or the sector area, and the angle or the radius is unknown. Write the same fraction statement as before, and solve it for the unknown.
To find the angle, find the fraction of the circle first. That fraction is the arc length divided by the whole circumference, or the sector area divided by the whole area.
An arc of $5 π$ cm on a circle of radius $9$ cm: the circumference is $18 π$ cm, so the fraction is $\frac{5 π}{18 π} = \frac{5}{18}$ . Then $θ = \frac{5}{18} \times 360 ° = 100 °$ .
A sector of area $15 π$ cm² in a circle of radius $6$ cm: the whole area is $36 π$ cm², so the fraction is $\frac{15}{36} = \frac{5}{12}$ . Then $θ = \frac{5}{12} \times 360 ° = 150 °$ .
To find the radius, put the letter $r$ into the fraction statement. A $40 °$ sector has area $20$ cm². Since $\frac{40}{360} = \frac{1}{9}$ :
$\frac{1}{9} \times π r^{2} = 20 \Rightarrow r^{2} = \frac{180}{π} = 57.29 \dots \Rightarrow r = 7.569 \dots = 7.57 cm$
Give an angle that is not exact to 1 decimal place, unless the question asks for something else. A radius or a length that is not exact goes to 3 significant figures.

Worked examples

Example 1. A circular pond has a diameter of $14$ m. Calculate (a) its circumference and (b) its area. Give each answer correct to 3 significant figures.

Show the working

Write down the radius first, because the area formula needs it: $r = 14 \div 2 = 7$ m.
(a) The diameter is given, so $C = π d$ is the quickest form.
$C = π \times 14 = 43.982 \dots$
To 3 significant figures that is $44.0$ m. The zero is the third significant figure, so write it.
(b) Square the radius, not the diameter.
$A = π \times 7^{2} = 49 π = 153.938 \dots$
So $A = 154$ m². If you had used $14$ , you would get $196 π = 616$ m², four times too big.

Answer: (a) $44.0$ m (b) $154$ m²

Example 2. The area of a circle is $20$ cm². Calculate its circumference. Give your answer correct to 3 significant figures.

Show the working

Find the radius first. Divide by $π$ , then take the square root.
$π r^{2} = 20 \Rightarrow r^{2} = \frac{20}{π} = 6.366 \dots \Rightarrow r = 2.5231 \dots$
Keep the full value of $r$ in the calculator. Do not write $2.52$ and carry on from it.
Now use the circumference formula with the full value.
$C = 2 π \times 2.5231 \dots = 15.853 \dots$
So $C = 15.9$ cm. If you had used the rounded $2.52$ , you would get $2 π \times 2.52 = 15.833 \dots = 15.8$ cm, which is wrong in the third significant figure.

Answer: $15.9$ cm

Example 3. Do not use a calculator. $O A B$ is a sector of a circle with centre $O$ and radius $9$ cm. Angle $A O B = 80 °$ . Find, in terms of $π$ , (a) the length of the arc $A B$ , (b) the area of the sector, (c) the perimeter of the sector.

Show the working

Find the fraction of the circle once, and simplify it so that the arithmetic is easy without a calculator.
$\frac{80}{360} = \frac{2}{9}$
(a) The whole circumference is $2 π \times 9 = 18 π$ cm. The arc is $\frac{2}{9}$ of it.
$\frac{2}{9} \times 18 π = 4 π$
(b) The whole area is $π \times 9^{2} = 81 π$ cm². The sector is $\frac{2}{9}$ of it.
$\frac{2}{9} \times 81 π = 18 π$
(c) The perimeter is the arc plus the two radii $O A$ and $O B$ : $4 π + 9 + 9 = 4 π + 18$ . Leave it as two terms. It is not $22 π$ .

Answer: (a) $4 π$ cm (b) $18 π$ cm² (c) $4 π + 18$ cm

Example 4. A sector of a circle of radius $8$ cm has an arc length of $15$ cm. (a) Calculate the angle of the sector. (b) Calculate the area of the major sector of the circle.

Show the working

(a) Write the arc as a fraction of the circumference, with $θ$ as the unknown.
$\frac{θ}{360} \times 2 π \times 8 = 15$
Multiply both sides by $360$ and divide both sides by $16 π$ . That undoes the fraction and the $2 π \times 8$ .
$θ = \frac{15 \times 360}{16 π} = 107.429 \dots$
An angle goes to 1 decimal place, so $θ = 107.4 °$ . This is the minor sector, because its angle is less than $180 °$ .
(b) The fraction of the circle in the minor sector is $\frac{15}{16 π}$ , the arc over the circumference. The same fraction of the area $64 π$ gives the minor sector exactly, with no rounding.
$\frac{15}{16 π} \times 64 π = 60$
The major sector is the whole circle minus the minor sector.
$64 π - 60 = 141.061 \dots$
Check another way: $\frac{360 - 107.429 \dots}{360} \times 64 π = 141.061 \dots$ as well. So the major sector is $141$ cm² to 3 significant figures.

Answer: (a) $107.4 °$ (b) $141$ cm²

Where people go wrong

Mistake: Putting the diameter into $π r^{2}$ : for a circle of diameter $10$ cm, writing $A = π \times 10^{2} = 100 π$ .

Instead: Halve the diameter first. $r = 5$ , so $A = 25 π$ cm².

The formula is built on the radius: the rearranged rectangle has height $r$ . Using the diameter doubles the length that is squared, so the area comes out four times too big.

Mistake: Squaring $π$ as well as the radius: for $r = 3$ , working out ${(π \times 3)}^{2} = 88.8$ .

Instead: Only $r$ is squared: $π r^{2} = π \times 9 = 28.3$ to 3 significant figures.

In $π r^{2}$ the power belongs to $r$ alone. Powers are done before multiplication, so square $3$ to get $9$ , then multiply by $π$ .

Mistake: Giving the arc length as the perimeter of a sector, or adding only one radius.

Instead: Perimeter of a sector = arc length + $2 r$ .

A sector has three edges: the curved arc and two straight radii. A perimeter is the whole way round, so all three edges count.

Mistake: Combining an exact perimeter such as $4 π + 18$ into $22 π$ .

Instead: Leave it as $4 π + 18$ . If a decimal is wanted, work out $4 π + 18 = 30.6$ to 3 significant figures.

Only like terms combine. $4 π$ is about $12.6$ , so $4 π + 18$ is about $30.6$ , but $22 π$ is about $69.1$ . They are different numbers.

Mistake: Using the marked minor angle when the question asks about the major sector.

Instead: Find the major angle first: $360 ° - θ$ . Or work out the minor sector and subtract it from the whole circle.

The fraction must describe the part you want. A $100 °$ minor angle gives $\frac{100}{360}$ of the circle, but the major sector is $\frac{260}{360}$ of it.

Check yourself

A circle has radius $4$ cm. Find its area in terms of $π$ .
Answer
$π \times 4^{2} = 16 π$ cm².
A circle has diameter $10$ cm. Calculate its circumference, correct to 3 significant figures.
Answer
$π \times 10 = 10 π = 31.415 \dots = 31.4$ cm.
The circumference of a circle is $50$ cm. Calculate its radius, correct to 3 significant figures.
Answer
$r = \frac{50}{2 π} = 7.957 \dots = 7.96$ cm.
Without a calculator, find the radius of a circle whose area is $64 π$ cm².
Answer
$π r^{2} = 64 π$ , so $r^{2} = 64$ and $r = 8$ cm.
A sector has angle $45 °$ and radius $8$ cm. Find, in terms of $π$ , its arc length and its area.
Answer
The fraction is $\frac{45}{360} = \frac{1}{8}$ . Arc length $= \frac{1}{8} \times 16 π = 2 π$ cm. Area $= \frac{1}{8} \times 64 π = 8 π$ cm².
Find the perimeter of the sector in the previous question, in terms of $π$ .
Answer
Arc plus two radii: $2 π + 8 + 8 = 2 π + 16$ cm.
In a circle of radius $3$ cm, the minor sector has angle $130 °$ . Find the area of the major sector in terms of $π$ .
Answer
The major angle is $360 ° - 130 ° = 230 °$ . Area $= \frac{230}{360} \times 9 π = \frac{23}{4} π$ cm², which is $5.75 π$ cm².
A sector of a circle of radius $5$ cm has area $10 π$ cm². Find the angle of the sector.
Answer
The whole area is $25 π$ , so the fraction is $\frac{10 π}{25 π} = \frac{2}{5}$ . The angle is $\frac{2}{5} \times 360 ° = 144 °$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Do not use a calculator in this question. Give each answer in terms of $π$ . A circle has centre $O$ and radius $6$ cm.

(a) Work out the circumference of the circle. 1 mark No calculator

Mark scheme

B1 12π

(b) Work out the area of the circle. 1 mark No calculator

Mark scheme

B1 36π

(c) A sector of this circle has an angle of $150 °$ at $O$ . Find the length of the arc of this sector. 2 marks No calculator

Mark scheme

M1 150/360 × 2 × π × 6 oe
A1 5π cao

(d) Find the perimeter of the major sector of the circle. 3 marks No calculator

Mark scheme

M1 210/360 × 2 × π × 6 oe, or 12π − their 5π
A1 major arc 7π soi
B1 final answer 7π + 12; FT their major arc + 12

Question 2. A circular rug has an area of $5$ m².

(a) Calculate the radius of the rug. 2 marks

Mark scheme

M1 πr² = 5 or r² = 5 ÷ π oe
A1 1.26 awrt (1.2615…)

(b) Calculate the circumference of the rug. 2 marks

Mark scheme

M1 2 × π × their r, or π × 2 × their r, or 2√(5π) oe
A1 7.93 awrt (7.926…), from the unrounded radius

(c) A second circular rug has a radius twice the radius of the first rug. Show that the area of the second rug is $20$ m². 2 marks

Mark scheme

M1 π × (2r)² oe, or π × (2 × 1.2615…)², or area scale factor 2² = 4 so 4 × 5
A1 4πr² = 4 × 5 = 20 shown, or 20.0 from the unrounded radius; a value from a rounded radius, e.g. 19.95, does not earn this mark

Question 3. $O A B$ is a sector of a circle with centre $O$ and radius $11$ cm. The length of the arc $A B$ is $16$ cm.

(a) Calculate angle $A O B$ . Give your answer correct to 1 decimal place. 3 marks

Mark scheme

M1 θ/360 × 2 × π × 11 = 16 oe
M1 θ = 16 × 360 ÷ (22π) oe; dep on the first M1
A1 83.3 awrt (83.33…)

(b) Calculate the area of the sector $O A B$ . 2 marks

Mark scheme

M1 their θ/360 × π × 11² oe
A1 88 or awrt 88.0

(c) Calculate the area of the major sector of the circle. 2 marks

Mark scheme

M1 π × 11² − their (b) oe
A1 292 awrt (292.1…)

How this is examined

Finding the angle of a sector when the arc length or sector area is given: A Calculate part that gives the radius and the arc length (or the sector area) and asks for the angle, to 1 decimal place A fraction statement with $θ$ in it, e.g. $\frac{θ}{360} \times 2 π \times 7 = 10$ , earns a method mark. Rearranging to $θ = \frac{10 \times 360}{14 π}$ earns a second. The angle to 1 decimal place, $81.9 °$ , earns the accuracy mark.
The perimeter of a sector is the arc plus two radii: A Find or Calculate part on a sector or a major sector, with the answer to 3 significant figures or in terms of π The correct angle for the sector asked about (for a major sector, $360 ° - θ$ ), then that fraction of $2 π r$ for the arc, then $+ 2 r$ for the two straight edges. In terms of π the answer has two terms, e.g. $7 π + 12$ , and it stays that way.
Arc length and sector area as a fraction of the whole circle, in terms of π: A short Work out or Find part on the non-calculator paper, with an angle such as $60 °$ , $120 °$ or $150 °$ that gives a simple fraction $\frac{θ}{360} \times 2 π r$ for an arc, or $\frac{θ}{360} \times π r^{2}$ for an area, written out with the numbers in, earns the method mark. The simplified multiple of $π$ , such as $5 π$ , with no decimal, earns the accuracy mark.
Working back from an area or a circumference to the radius: A Calculate part that gives the area (or circumference) of a circle and asks for the radius, and may then ask for something that uses that radius The equation $π r^{2} = A$ (or $2 π r = C$ ) with the given value in, then $r$ correct to 3 significant figures. In the next part, use the full calculator value of $r$ , not the rounded one, so that your final answer is accurate.
Circumference and area from a radius or a diameter: A Calculate or Work out part on a circular object, with the answer to 3 significant figures or in terms of π The radius, not the diameter, in $π r^{2}$ ; $π d$ or $2 π r$ for the circumference; the answer to 3 significant figures (e.g. $44.0$ , not $44$ ) or as a simplified multiple of $π$ ; and the correct unit, cm for a length and cm² for an area.

Words to know

Radius: The distance from the centre of a circle to its edge, written $r$ .
Diameter: A straight line across a circle through its centre. Its length is twice the radius: $d = 2 r$ .
Circumference: The distance all the way round a circle: $C = π d = 2 π r$ .
π (pi): The number you get when you divide the circumference of any circle by its diameter: $3.14159 \dots$ , with digits that never end or repeat.
In terms of π: An exact answer with $π$ left as a symbol and the number in front simplified, e.g. $36 π$ cm² or $7 π + 12$ cm.
Arc: Part of the circumference of a circle. The minor arc is the shorter part between two points; the major arc is the longer part.
Sector: The region of a circle between two radii and the arc joining their ends. Its area is $\frac{θ}{360} \times π r^{2}$ .
Minor sector and major sector: The two sectors made by two radii. The minor sector has an angle less than $180 °$ ; the major sector has an angle more than $180 °$ . Their angles add up to $360 °$ .
Perimeter of a sector: The arc length plus the two radii: arc length $+ 2 r$ .

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