Circles, Arcs and Sectors

What page 2 gives you, and what it does not

The syllabus prints a List of formulas on page 2 of every question paper, and for this topic it gives you exactly two formulas: the area of a circle, $A=\pi r^2$, and the circumference, $C=2\pi r$. The same two are printed for the Core papers (Paper 1 and Paper 3) and for the Extended papers (Paper 2 and Paper 4).

Nothing about part of a circle is printed. There is no arc length formula and no sector area formula on the sheet, so the only thing you can do is remember that a sector is a fraction of the whole circle. That single idea is worth more marks in this topic than any other.

The syllabus is explicit that the sheet is not complete: it says not all required formulas are given, and that the Notes column of the subject content shows where a formula is given and where knowledge of it is required. A question about a sector is testing knowledge you have to supply yourself.

Circumference: the distance round the circle

The circumference is the perimeter of a circle. It is $C=2\pi r$ in terms of the radius, or $C=\pi d$ in terms of the diameter, and the two are the same statement because $d=2r$. Use $C=\pi d$ whenever the question hands you a diameter: it saves a line of working and one chance of halving the wrong number.

$\pi$ is a fixed number, about 3.142, not a variable. You never work $\pi$ out and you never round it in the middle of a calculation - use the $\pi$ button, and round once at the end.

When a question says give your answer in terms of $\pi$, leave the symbol in. Radius 9 cm gives a circumference of $18\pi$ cm, and that exact answer is what scores. Only turn it into a decimal if you are asked for one, and then round to the accuracy the question asks for, usually 3 significant figures.

• Diameter 14 cm: $C=\pi \times 14=14\pi$ cm, which is 44.0 cm to 3 significant figures
• Radius 9 cm: $C=2\pi \times 9=18\pi$ cm, which is 56.5 cm to 3 significant figures
• A circumference of $20\pi$ cm means the diameter is 20 cm, because $C=\pi d$

Area of a circle, and the radius-not-diameter trap

The area of a circle is $A=\pi r^2$. The radius is what gets squared, and $\pi r^2$ means $\pi \times r\times r$. A circle of radius 9 cm has area $\pi \times 81=81\pi$ cm2, which is 254 cm2 to 3 significant figures.

If the question gives a diameter, halve it first. A circle of diameter 14 cm has radius 7 cm, so its area is $\pi \times 49=49\pi$ cm2, about 154 cm2. Squaring the 14 gives $196\pi$ cm2, four times too big, and the mark scheme is written to catch exactly that.

Area answers are in square units: cm2, m2, mm2. An area written in cm is a lost mark even when the number is right, and the same is true the other way round for a length.

Arc length: a fraction of the circumference

An arc is part of the circumference, and a sector is the slice of the circle cut off by two radii. The angle at the centre tells you what fraction of the whole circle you have: ${90}^{\circ }$ is $\frac{90}{360}=\frac{1}{4}$ of a turn, ${120}^{\circ }$ is $\frac{1}{3}$, and ${45}^{\circ }$ is $\frac{1}{8}$.

So the arc length is that fraction of the circumference: arc length $=\frac{\theta }{360}\times 2\pi r$. With $r=12$ cm and $\theta ={75}^{\circ }$ the fraction is $\frac{75}{360}=\frac{5}{24}$, so the arc is $\frac{5}{24}\times 24\pi =5\pi$ cm, about 15.7 cm.

The Core syllabus says the sector angle is a factor of 360, which is what keeps the fraction tidy: 30, 36, 40, 45, 60, 72, 90, 120 and 180 all give a simple fraction. Extended candidates also meet the major sector, where the angle is bigger than ${180}^{\circ }$ and the fraction is more than one half.

• Arc length $=\frac{\theta }{360}\times 2\pi r$, or $\frac{\theta }{360}\times \pi d$ if you have the diameter
• ${180}^{\circ }$ is half the circumference, $\pi r$
• ${90}^{\circ }$ is a quarter of the circumference, $\frac{\pi r}{2}$

Sector area: the same fraction of the area

Sector area $=\frac{\theta }{360}\times \pi r^2$. The fraction is exactly the one you used for the arc, because a sector is the same slice of the circle either way. Radius 10 cm with an angle of 126 degrees gives $\frac{126}{360}=0.35$, so the area is $0.35\times 100\pi =35\pi$ cm2, about 110 cm2.

Look at what is squared and what is not. The radius is squared; the fraction is not, and the angle is not. Squaring the fraction as well is the classic error here and it makes the answer about a third of the true value.

Keep the two formulas apart by their units. The arc is a fraction of the circumference, so it is a length in cm; the sector area is a fraction of the area, so it is in cm2. If you end up with a length where an area was asked for, you have used the wrong fraction of the wrong formula.

Working backwards: to the angle, or to the radius

The two formulas reverse. A sector of radius 12 cm has area $30\pi$ cm2, and the whole circle is $144\pi$ cm2, so the sector is $\frac{30}{144}=\frac{5}{24}$ of the circle and the angle is $\frac{5}{24}\times 360={75}^{\circ }$. The fraction is the bridge between the part and the whole.

To find a radius, set the formula equal to what you have been given and solve. An angle of ${90}^{\circ }$ with an area of $100\pi$ cm2 gives $\frac{1}{4}\pi r^2=100\pi$, so $r^2=400$ and $r=20$ cm. When the data is in terms of $\pi$, the $\pi$ cancels from both sides, which is why these questions are usually cleaner than they look.

Check the answer against the picture before you move on. An angle bigger than ${360}^{\circ }$, a fraction bigger than 1, or an arc longer than the whole circumference all mean the fraction has been written upside down.

• Fraction of the circle: sector area divided by $\pi r^2$, or arc length divided by $2\pi r$
• Angle $=360\times$ the fraction
• Given ${90}^{\circ }$ and $100\pi$ cm2, $r^2=400$ so $r=20$ cm