Area of Circles and Sectors

Cut a circle into thin slices and lay them alternately point-up and point-down, and they push together into something very close to a rectangle. Its height is the radius, and its length is half the circumference, which is πr. So the area is πr times r, giving A = πr². The critical reading of that formula is π times (r times r): you square the radius first and then multiply by π. It does not mean (πr) squared — that's an order-of-operations question, and powers come before multiplication.

Area only ever uses the radius. So when a question gives you the diameter, halve it before you do anything else. For a diameter of 12, the radius is 6, and the area is π times 36, about 113.1 square centimetres. Using 12 by mistake gives 452.4 — four times too big, because squaring the wrong number squares the mistake as well. This is worth more care than the circumference formula, where either measurement works. Get into the habit of writing 'r = …' on your page before you touch the calculator.

A sector is a fraction of the whole circle, and it's the same fraction you used for arc length — the centre angle over 360. So find the whole circle's area first, then take that fraction of it. For a 120-degree sector with radius 9: the full circle is π times 81, about 254.5, and 120 over 360 is a third, so the sector is about 84.8 square centimetres. Written as a formula, sector area = (θ over 360) × πr². A half circle is ½πr² and a quarter is ¼πr², which are just the same rule with the fraction already worked out.

A ring — an annulus — is a big circle with a smaller one removed. Find each area in full, then subtract: π times 100 is 314.2, π times 36 is 113.1, and the ring is about 201.1 square centimetres. The tempting shortcut is to subtract the radii first and square the result, but π times (10 − 6) squared gives 50.3, which isn't close. Squaring and subtracting can't be swapped around. The same 'whole minus hole' method handles a circle cut from a square, or a path laid around a pond: compute each piece completely, and only then take one away from the other.

Two ways questions finish. If asked to leave the answer in terms of π, square the radius but keep the symbol: for r = 4, the area is 16π square centimetres. And working backwards means undoing the formula in reverse order — the formula squares last, so you divide by π first and take the square root last. Given an area of 50, r² is 50 ÷ π, about 15.9, so r is about 4.0 cm; no ± is needed because a radius is positive by definition. This is also where circle area gets genuinely useful. 'Which pizza is better value?' is an area-per-dollar question, and it surprises people that doubling the radius makes the area four times bigger rather than twice.