Circle Calculator

Solve for radius, diameter, circumference, or area. Includes sector arc calculations and student proofs.

Select Input Type

Enter radius value

Sector & Arc Calculator

Central Angle

Sector Area:19.635

Arc Length:7.85398

Chord Length:7.07107

Visual Dimension

Radius (r)5

Diameter (d)10

Circumference (C)31.4159

Area (A)78.5398²

Step-by-Step Proof

1. Radius is given directly: r = 5

2. Diameter (d) = 2 * r = 2 * 5 = 10

3. Circumference (C) = 2 * π * r = 2 * 3.14159 * 5 = 31.4159

4. Area (A) = π * r² = 3.14159 * 5² = 78.5398

Circle geometry rules and formulas

The geometric proportions of a circle are linked via simple algebraic equations using the constant $\pi \approx 3.14159$. The radius $r$ forms the core dimension:d = 2r, C = 2\pi r, and A = \pi r^2.

Sector and arc segments are calculated by dividing the central angle $\theta$ against a full revolution ($360^\circ$ or $2\pi$ radians). Arc length is defined by $L = r\theta$, and sector area equals $A_s = \frac12 r^2 \theta$.

Built and maintained by Meet Shah · Last updated 1 August 2026

What this tool is used for

Getting every circle property from any single one of them.
Working out an area from a measured circumference.
Finding a radius from a diameter you can measure but cannot halve accurately.
Seeing the formula alongside the result for a lesson.
Checking a stated area against the dimension given.

Frequently Asked Questions

Which measurements can it solve from?: Any one of radius, diameter, circumference or area — each determines the other three, because they are all expressions in a single parameter. Give it the area and it works backwards through r = √(A/π) rather than requiring you to start from the radius.
Why is π irrational and does it matter here?: π has no exact decimal or fractional representation, proved irrational by Lambert in 1761, so every numeric circle answer is a truncation. It matters only at the precision boundary: 15 digits is enough to compute the circumference of the observable universe to within an atom's width.
How does area scale with radius?: As the square, which is the fact people underestimate most. Doubling a pizza's diameter quadruples the food — a 16-inch pizza is 78% more than a 12-inch, not 33% more. The same relationship makes a modest increase in pipe diameter a large increase in flow.
What is the difference between an arc and a chord?: An arc is the curved distance along the circumference; a chord is the straight line between the same two endpoints. The chord is always shorter, and the gap grows with the angle — at 180° the chord is the diameter while the arc is half the circumference, about 57% longer.
Why is a circle the most efficient shape for enclosing area?: Because it maximises area for a given perimeter — the isoperimetric inequality, and the reason bubbles, planets and pressure vessels are round. A square with the same perimeter encloses about 21% less, which is why circular is the default whenever material cost scales with the boundary.

Common errors and gotchas

Using a diameter where the formula wants a radius, which is a factor of four out on area.
Rounding pi early, which propagates into every derived value.
Confusing circumference with area, which have different units.
Forgetting area scales with the square of the radius, so doubling it quadruples the area.
Mixing units between the input and the expected output.