Circumference and Arc Length

Take any circle and roll it along a line for one full turn. The distance it covers is always a little over three diameters — and that 'little over three' is exactly the same number for every circle, big or small. We call it π, about 3.14159. So the circumference divided by the diameter is always π, which gives the formula C = πd. Since the diameter is twice the radius, the same formula can be written C = 2πr. Both say the identical thing; you pick whichever matches the measurement you've been given.

Almost every lost mark in this topic comes from mixing up radius and diameter. If a circle has radius 7 cm, then C = 2 × π × 7, which is about 44 cm. Using π × 7 instead gives 22 — exactly half, and completely wrong. So before you touch the calculator, write down which measurement you've been given. Radius means use 2πr; diameter means use πd. And remember the two are linked: d = 2r, and r = d ÷ 2. Getting this backwards doubles or halves your answer, so it's worth ten seconds of care every time.

An arc is just part of the circumference, so the whole method is working out what fraction of the circle you have. The angle at the centre tells you: 90 degrees out of 360 is a quarter, so a 90-degree arc on a radius-8 circle is a quarter of 2π times 8, about 12.6 cm. The general rule is arc length = θ over 360, times 2πr, where θ is the centre angle. Learn that fraction and you never need separate rules for half circles and quarter circles — 180 over 360 is a half, 120 over 360 is a third. And sanity-check every answer: an arc must be shorter than the full circumference.

A sector — a pizza slice — has three edges: two straight radii and one curved arc. So its perimeter is the arc plus r plus r, which is (θ over 360) times 2πr, plus 2r. For a 120-degree sector with radius 9, the arc is about 18.85 and the two radii add 18, giving 36.85 cm. The thing to watch is that 'arc length' and 'perimeter of the sector' are different questions with different answers: the arc is the curve alone, while the perimeter includes the two straight sides. A semicircle works the same way — its curve is πr, and its perimeter adds the diameter to give πr + 2r.

Two final things questions like to ask. First, 'leave your answer in terms of π' means don't multiply out: for a radius of 6, the answer is 12π cm, and writing 37.7 would actually lose the mark. Second, working backwards. C = πd is an equation, so given the circumference you divide instead of multiplying: if C is 50, then d = 50 ÷ π, about 15.9 cm — and if the question wanted the radius, halve that at the end. This is genuinely useful: a wheel of diameter d travels πd in one full turn, so a bike wheel 0.7 m across covers about 2.2 m per revolution, meaning 100 metres takes roughly 45 turns.