Circles, arcs and sectors

The parts of a circle

Every point on a circle is the same distance from its centre. That distance is the radius, $r$. A diameter is a straight line from one side of the circle to the other through the centre, so the diameter is twice the radius: $d=2r$.

The distance all the way round the circle is its circumference. Part of the circumference is an arc. A sector is the slice of the circle between two radii, like a slice of pizza; the angle between the two radii, at the centre, is the sector angle.

Read every question carefully to see whether you have been given the radius or the diameter. Using a diameter in place of a radius is the most common way to lose marks in this topic.

A radius, a diameter, an arc and a sector.

Circumference

For every circle, however large or small, the circumference divided by the diameter gives the same number, called $\pi$ (pi). $\pi =3.14159\dots$; its decimals never end or repeat. So the circumference is $\pi$ times the diameter:

$$C=\pi d=2\pi r$$

The formula $C=2\pi r$ is printed in the List of formulas on the exam paper. Use the $\pi$ key on your calculator, not 3.14, so that your answer is accurate to 3 significant figures.

For example, a circle of radius 4.5 cm has circumference $2\times \pi \times 4.5=28.274\dots =28.3$ cm.

Area of a circle

$$A=\pi r^2$$

This formula is also printed on the paper. Square the radius first, then multiply by $\pi$: $\pi r^2$ means $\pi \times r\times r$, not $(\pi r{)}^2$.

Why it works. Cut a circle into many thin sectors and lay them side by side, alternately point up and point down. They make a shape that is almost a rectangle. Its height is the radius $r$, and its length is half the circumference, $\pi r$, because half the arcs are along the top and half along the bottom. So the area is $\pi r\times r=\pi r^2$. The more sectors you use, the closer the shape is to a true rectangle.

Area is in square units. A circle of radius 4.5 cm has area $\pi \times {4.5}^2=63.617\dots =63.6{\text{cm}}^2$.

Exact answers in terms of π

On the non-calculator papers, and whenever a question says "give your answer in terms of $\pi$", leave $\pi$ as a symbol. This gives the exact answer.

• A circle of radius 6 cm: circumference $=2\times \pi \times 6=12\pi$ cm; area $=\pi \times 6^2=36\pi {\text{cm}}^2$.
• Write the number in front of $\pi$: $12\pi$, not $\pi 12$. Simplify fractions: $\frac{36\pi }{4}=9\pi$.
• A perimeter may have a $\pi$ part and a whole-number part, such as $3\pi +12$. These cannot be combined, so leave the answer in that form.

Do not change an answer in terms of $\pi$ into a decimal: $36\pi$ is exact, while $113.1$ is only approximate and loses the mark.

Working backwards to a radius or an angle

If you know the circumference or the area, rearrange the formula to find the radius.

• From a circumference: $C=2\pi r$, so $r=\frac{C}{2\pi }$.
• From an area: $A=\pi r^2$, so $r^2=\frac{A}{\pi }$ and $r=\sqrt{\frac{A}{\pi }}$.

Do not forget the square root in the second case. Then use the radius to find whatever else is asked for, keeping the full calculator value (not the rounded one) for the next step.

In terms of $\pi$ this is often very quick: if the circumference is $18\pi$ cm, then $2\pi r=18\pi$, so $r=9$ cm.

Arcs and sectors as fractions of a circle

A sector is a fraction of a whole circle, and the fraction is fixed by its angle: the whole circle has ${360}^{°}$ at the centre, so a sector with angle $\theta$ is $\frac{\theta }{360}$ of the circle. Its arc is the same fraction of the circumference, and its area is the same fraction of the area.

In the Core papers the sector angle is a factor of 360, so the fraction is simple:

• ${180}^{°}$ is $\frac{1}{2}$ (a semicircle), ${90}^{°}$ is $\frac{1}{4}$ (a quarter circle), ${120}^{°}$ is $\frac{1}{3}$;
• ${60}^{°}$ is $\frac{1}{6}$, ${45}^{°}$ is $\frac{1}{8}$, ${40}^{°}$ is $\frac{1}{9}$, ${30}^{°}$ is $\frac{1}{12}$.

So for a sector:

• arc length $=$ fraction $\times 2\pi r$;
• sector area $=$ fraction $\times \pi r^2$;
• perimeter of the sector $=$ arc length $+2r$, because its edge is the arc and two radii.

The perimeter of a semicircle of radius $r$ is therefore $\pi r+2r$: half the circumference plus the diameter.

A sector with angle $\theta$ is $\frac{\theta }{360}$ of the whole circle.