Circle calculator

Every measurement of a circle follows from any other, because they are all tied together by a single constant.

The two formulas

Everything about a circle comes from two relationships:

Circumference — C = 2πr, equivalently πd
Area — A = πr²

π is the ratio of any circle's circumference to its diameter, the same for every circle regardless of size — roughly 3.14159. That constancy is the whole reason a single pair of formulas covers all circles.

Working backwards

The calculator accepts any of the four measurements because each rearranges cleanly:

From area: r = √(A/π)
From circumference: r = C/(2π)
From diameter: r = d/2

Going from area to radius is the one people get wrong most often, usually by forgetting the square root or dividing by 2π out of habit.

Why area scales with the square

Double the radius and the circumference doubles, but the area quadruples. A 16-inch pizza has not twice the food of an 8-inch one — it has four times as much. This catches people out constantly in pricing, in pipe capacity, and in comparing round tables.

The same relationship works in reverse: to double the area of a circle you multiply the radius by √2, about 1.414, not by 2.

Sectors, arcs and chords

A sector is a wedge. Since a full circle is 360°, a sector of angle θ is simply that fraction of the whole:

Arc length = C × θ/360
Sector area = A × θ/360
Chord = 2r·sin(θ/2) — the straight line across, not the curve

A quarter circle at 90° therefore has a quarter of the area and a quarter of the circumference as its arc, which is a useful sanity check on the output.

Exact answers

The calculator shows results in terms of π as well as decimals, because most homework wants the exact form. A circle of radius 5 has area 25π, which is precise; 78.5398 is a rounded approximation. In engineering the decimal is what you want, but in algebra keeping π symbolic avoids compounding rounding error through later steps.

Common questions

How do I find the radius from the area?

Divide the area by pi, then take the square root: r = sqrt(A/pi). Forgetting the square root is the most common error here.

Why does doubling the radius quadruple the area?

Because area depends on the radius squared. Doubling r gives (2r) squared = 4r squared. This is why a 16-inch pizza contains four times as much as an 8-inch one, not twice.

What is the difference between an arc and a chord?

An arc is the curved portion of the circumference between two points. A chord is the straight line joining them. The arc is always longer, and the chord is 2r sin(theta/2).

Should I leave answers in terms of pi?

For algebra and exams, usually yes — 25 pi is exact while 78.54 is rounded. For anything you are physically building, the decimal is what you need.