Radius of a Circle Calculator

How to use the radius of a circle calculator

What this calculator does

This calculator finds the radius of a circle when you know its circumference, area, or diameter. It also synchronizes all four standard circle measurements so you can move between them without manually rearranging formulas. The result is an exact geometric identity subject only to display rounding; it does not account for measurement uncertainty, material thickness, an irregular boundary, or a shape that is only approximately circular. OpenStax's explanation of circle circumference and the relationship between radius and diameter provides a useful conceptual reference.

When to use it

Use the tool when measuring a round tabletop from edge-to-edge, converting a wheel's circumference into its radius, finding the radius of a circular garden from its area, or checking whether several recorded circle measurements are consistent. It is also useful for classroom work, fabrication estimates, design sketches, and any task where the radius is needed before calculating arc length, sector area, or other circle properties.

How to calculate

1. The calculator opens with a complete demonstration: a 10 cm diameter produces a 5 cm radius, a 31.4159 cm circumference, and a 78.5398 cm² area. The Excel workbook is ready immediately.
2. Edit Circumference, Area, or Diameter. The field you most recently change becomes the source measurement, and the other two inputs are recalculated automatically.
3. Select a Measurement unit when needed. The calculator converts the existing length and area values; it does not merely replace the unit label.
4. Read the primary Radius, the supporting results, and the circle relationship table. Use Download Excel to save the current canonical values and formulas in a validated .xlsx workbook.
5. Use Reset to clear the demonstration and all calculated content. Reset intentionally leaves the data fields empty and disables Excel export until you enter a complete valid measurement again.

Input guide

Circumference is the distance around the circle. It is optional as an entry method because you may use another measurement instead, but any value entered must be a positive finite number in the selected length unit. A realistic example is 31.4159 cm. A larger circumference produces a proportionally larger radius. Enter decimal points with a period; commas are accepted only in valid thousands groupings, so “1,500.5” is valid while decimal-comma “1,5” is rejected rather than silently interpreted. Scientific notation such as “1e3” is accepted when useful.

Area is the region enclosed by the circle and uses the square of the selected unit. It must be positive and finite. For example, 78.5398 cm² corresponds to a 5 cm radius. Because radius is the square root of area divided by π, quadrupling the area doubles the radius. A common mistake is entering a linear measurement into this field or forgetting that square units change by the square of the conversion factor.

Diameter is the straight-line distance through the center from one side of the circle to the other. It must be positive and finite; 10 cm is the startup example. The radius is always half the diameter, so increasing diameter by 20% also increases radius by 20%. Do not confuse diameter with a chord that misses the center.

Measurement unit is required and supports millimeters, centimeters, meters, inches, feet, and yards. Length outputs use that unit, while area uses its squared form. Changing from centimeters to meters converts 5 cm to 0.05 m and 78.5398 cm² to approximately 0.007854 m². The unit selector does not mix systems within one calculation.

Output guide

Radius is the distance from the center to the circle's edge and is the primary exact identity produced by the selected source formula. Radius formula identifies the active rearrangement: r = d ÷ 2, r = C ÷ (2π), or r = √(A ÷ π). Diameter, Circumference, and Area are equivalent measurements derived from that same radius. A zero or negative result is not meaningful for this calculator, so those inputs are rejected. Very high values remain estimates only to the extent that the original physical measurement is approximate.

The pills labeled Based on, Length unit, Area unit, and π precision summarize the current source and units. Circumference ÷ diameter is a consistency check and should equal π apart from display rounding. In the Circle relationship table, “Measurement” names the quantity, “Current value” shows its synchronized value, “Relationship” states the equation used to recover radius, and “Radius recovered” confirms that every row describes the same circle.

Worked example

Start with the demonstrated diameter d = 10 cm. The diameter formula gives r = d ÷ 2 = 10 ÷ 2 = 5 cm. From that radius, circumference is C = 2πr = 2 × π × 5 = 31.4159265359 cm, displayed as 31.4159 cm. Area is A = πr² = π × 5² = 78.5398163397 cm², displayed as 78.5398 cm². These are the same values shown in the initial controls, result cards, relationship table, and downloadable workbook.