A circle has no straight sides, so it is measured with a special number, π (pi) ≈ 3.14. Pi is the same for every circle in the universe: it links how far across a circle is to how far it is around. From wheels and clock faces to pizzas and plant pots, two formulas cover almost everything.
Why this matters: anything round is a circle problem. How much fencing rings a circular pond (circumference), how much pizza you actually get (area), how far a wheel rolls in one turn (also circumference) — all come from πd and πr². Pi is one of the most useful constants in all of maths and engineering, and these two formulas are your entry to it.
Radius r (centre to edge); diameter d goes right across (= 2r)
The two key formulas
Circumference is the distance right around the edge.
Here r is the radius — the distance from the centre out to the edge — and d is the diameter, the full distance right across through the centre. The diameter is always twice the radius (d = 2r), which is why πd and 2πr are the same thing written two ways.
Area fills the inside: square the radius first, then × π.
Want
Formula
Note
Circumference
πd or 2πr
uses diameter
Area
πr²
square the radius first
Radius from d
r = d ÷ 2
half the diameter
Diameter from C
d = C ÷ π
work backwards
Common mistake: mixing up the two formulas, and squaring in the wrong order. Circumference uses the diameter (πd); area uses the radius squared (πr²). And πr² means square the radius first, then multiply by π — never multiply π and r and then square.
Arcs and sectors
A sector is a pizza slice; its fraction is angle ÷ 360.
An arc is just part of the circumference (a curved slice of the edge); a sector is a "pizza slice" bounded by two radii. To find either, work out what fraction of the whole circle your slice is by comparing its angle to 360°, then take that fraction of the full circumference or area.
Fraction of circle = angle ÷ 360
A 90° sector is 90 ÷ 360 = ¼ of the whole circle — so its area is a quarter of πr². A 180° sector (a semicircle) is half.
Semicircle, radius 10 cm (π = 3.14): full area = 3.14 × 100 = 314, so the semicircle is ½ × 314 = 157 cm².
Memory hook: "Cherry pie's delicious" for Circumference = π × d, and "Apple pies are too" for Area = π × r². For any slice, multiply by (angle ÷ 360).