Circles

The parts of a circle

Before any formula, the vocabulary — most mistakes here are really vocabulary mistakes:

• Centre — the middle point.
• Radius ($r$) — from the centre to the edge.
• Diameter ($d$) — right across, through the centre. Twice the radius.
• Circumference ($C$) — the distance all the way round the outside.
• Chord — a straight line joining two points on the edge, not through the centre.
• Arc — part of the circumference.

Where $\pi$ comes from

Measure the circumference of any circular object and divide by its diameter. A tin, a plate, a coin — you always get the same number, roughly $3.14$. That constant ratio is $\pi$.

So $C \div d = \pi$, which rearranges to $C = \pi d$. And since $d = 2r$, the same formula can be written $C = 2\pi r$.

Your textbook uses $\pi \approx 3.14$, which is fine for this course. Your calculator’s $\pi$ button is more accurate, and your answers will differ very slightly — say which you used.

The two formulas, and the trap between them

$C = 2\pi r \qquad\qquad A = \pi r^2$

Almost every lost mark in this topic is one of these three:

1. Using the diameter in the area formula. $A = \pi r^2$ needs the radius. If you are given $d = 10$, use $r = 5$, not 10. Getting this wrong multiplies your answer by four.
2. Squaring the wrong thing. $\pi r^2$ means $\pi \times r \times r$. Square the radius first, then multiply by $\pi$.
3. The wrong units. Lengths are cm; areas are $\text{cm}^2$.

Working backwards

Questions often give you the area or circumference and ask for the radius. Just reverse the operations:

For the area one, divide by $\pi$ first, then square root. Doing it the other way round gives nonsense.

Semicircles and compound shapes

For area, halve the circle’s area — straightforward.

For perimeter, halve the curved part, then add the straight diameter back on. A perimeter is the distance round the whole edge, and the flat side is part of that edge. Forgetting it is the classic semicircle error.

For a shape built from a rectangle and a semicircle, work out each piece separately and combine them — and think carefully about which edges are on the outside of the finished shape.