Circles: Circumference & Area

The radius r goes from the centre to the edge. The diameter d goes all the way across, so d = 2r.

Circumference (the distance around) is C = 2πr, or πd.
Area (the space inside) is A = πr².

π ≈ 3.14159, and it is the same for every circle: the circumference divided by the diameter always gives π.

Which formula is which? The squared one gives area, because area always involves two dimensions multiplied. If your answer has r², it is an area.

π is defined as the ratio C/d, constant for all circles by similarity — all circles are similar figures, so corresponding measurements share a fixed ratio. From that definition C = πd = 2πr follows immediately.

The area formula can be seen by cutting a circle into many thin sectors and interleaving them into an approximate rectangle of height r and width ½C = πr, giving A = πr² in the limit. That construction is also the intuition behind the derivative relationship dA/dr = 2πr = C, which is why the circumference is exactly the rate at which area grows as the radius increases.

Arc length and sector area are proportional slices of the whole: an angle of θ degrees gives arc (θ/360)·2πr and sector area (θ/360)·πr². Answers are usually left in terms of π unless a decimal is requested, since 12π is exact where 37.699 is not.