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Calcimator

Circle Calculator

Calculate radius, diameter, circumference, area, sector area, and arc length of a circle from any known value.

About this calculator

This calculator lets you enter any one of a circle's four basic measurements — radius, diameter, circumference, or area — and works backward to recover the radius before computing every other figure from that single number. Diameter divides by two to reach the radius; circumference divides by two pi; area is inverted through a square root, since area equals pi times radius squared. Once the radius is known, everything downstream, including Area and Circumference themselves, is recalculated fresh from it rather than simply echoing whatever value was typed in, so switching Known Value from Area to Radius and back produces the same set of numbers either way, modulo the rounding each field applies.

The Sector Angle field controls two outputs only — Sector Area and Arc Length, both scaled down from the full circle's totals by the fraction the angle represents of 360 degrees — and it has no bearing whatsoever on the whole-circle Area, Circumference, Radius, or Diameter figures; a sector angle of 10 degrees and one of 350 degrees produce identical values for those four outputs while the sector-specific pair scales dramatically. This is a genuinely closed system: nothing here handles ellipses or any shape other than a true circle, and the tool does not account for a partial circle described any way other than through the Sector Angle field, such as a chord length or a segment area.

Inputs

°

Results

Area

78.54

Circumference31.42
Radius5
Diameter10
Sector Area78.54
Arc Length31.42

Area of a circle

  1. 1.Square the radius
    r2=52=25

    Area scales with the square of a length, not the length itself — doubling the radius quadruples the area. This step is where that relationship enters.

  2. 2.Multiply by π
    A=π×25=78.54

    π is the fixed ratio between a circle's circumference and its diameter. It converts the squared radius — the area of a square on the radius — into the area of the circle itself.

Archimedes proved that a circle's area equals that of a right triangle whose height is the radius and whose base is the circumference, and bounded π between 3+10/71 and 3+1/7 by comparing inscribed and circumscribed 96-gons.
Archimedes, Measurement of a Circle, Propositions 1 and 3·c. 250 BCE
How to Use This Calculator
  1. Enter any single known circle measurement — radius, diameter, circumference, or area — and all others are derived.
  2. Key relationships: diameter = 2r; circumference = 2πr; area = πr².
  3. For a circular sector, enter the central angle (in degrees) to get sector area and arc length.
  4. Sector area = (θ/360) × πr²; arc length = (θ/360) × 2πr.
  5. The pie chart visualizes the sector vs. remaining arc, which is useful for checking proportional reasoning.
  6. All inputs must be positive; radius and diameter inputs override each other — enter only one at a time.

How the result changes with Value

ValueArea
2.519.64
3.7544.18
7.5176.71
13530.93

What each input means

Known Value
Calculation mode to use.
Value
The input value.

How this is calculated

Formula

A = πr², C = 2πr

Worked example, using the default values

  1. Identify Input Parameters
    Known Value = 0, Value = 5, Sector Angle (degrees) = 360 = 3 input(s) provided
  2. Calculate Area
    Area
    78.5398 = 78.5398
  3. Calculate Circumference
    Circumference
    31.4159 = 31.4159
  4. Calculate Radius
    Radius
    5 = 5

Engine last updated . Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.

Frequently Asked Questions

Does the Sector Angle field change the circle's total Area or Circumference?

No. Sector Angle only feeds into Sector Area and Arc Length, both of which scale the whole-circle totals down by the fraction of 360 degrees the angle represents. The whole-circle Area, Circumference, Radius, and Diameter outputs are computed the same way no matter what Sector Angle is set to.

How does the calculator find the radius when I enter Area instead?

It inverts the area formula by dividing the entered value by pi and taking the square root, since area equals pi times radius squared. Once that radius is recovered, every other output — diameter, circumference, sector figures — is computed fresh from it rather than from the area you originally typed.

If I switch Known Value from Area to Circumference, will I get the same circle back?

Yes, aside from small rounding differences. Both paths solve for the same radius first — one by taking a square root, the other by dividing by two pi — and every output after that point is derived identically regardless of which measurement you started from.

Can this calculator handle an ellipse or a circle segment?

No. Every formula here assumes a true circle, and the only way to describe a partial shape is through the Sector Angle field, which measures a pie-slice-style sector. A chord length or a segment area — the region cut off by a straight line rather than two radii — is not something this tool computes.

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