Geometry & Shapes
Find a Circle’s Radius, Diameter, Area and Circumference
Recover a circle’s dimensions from one known measurement, verify inverse formulas, and distinguish a circle from a circular segment.
Identify the known circle measurement
For a true circle, one positive radius determines its diameter, circumference and area. The radius r runs from the centre to the boundary, while the diameter d crosses the centre and equals 2r. Circumference C is the boundary length and area A is the enclosed region. The basic formulas are C=2πr=πd and A=πr². Name the known quantity before calculating. A number described only as the size of a circle is ambiguous because it might mean diameter, area or circumference, with different units and formulas.
Start from a measured radius
For radius 5 centimetres, the diameter is 10 centimetres. Circumference is 10π centimetres, approximately 31.4159 centimetres, and area is 25π square centimetres, approximately 78.5398 square centimetres. Keep π symbolic until you need a decimal approximation. Replacing it with 3.14 early produces slightly different rounded results. The length units remain centimetres for radius, diameter and circumference; the area uses square centimetres. A label of 78.54 centimetres would describe the wrong kind of quantity even if the digits came from the correct multiplication.
Recover the radius from diameter or circumference
From a diameter, divide by two: r=d/2. From circumference, divide by 2π: r=C/(2π). If the circumference is 20π metres, the radius is 10 metres and the diameter is 20 metres. If it is measured as 62.8 metres, the inferred radius is approximately 9.9949 metres using full π. That small difference reflects the rounded measurement rather than a contradiction. Do not divide a circumference by two and call the result the radius; that ignores the circular relationship expressed by π.
Recover dimensions from area
From area, use r=√(A/π). For area 100π square metres, divide by π to obtain 100 square metres, then take the positive square root to obtain radius 10 metres. The diameter is 20 metres and the circumference is 20π metres. The positive root is appropriate because a geometric radius is a nonnegative length. An area must not be inserted directly into a radius field. Squaring it again would produce fourth-power units rather than the desired area or length.
Convert units at the correct power
A circle of radius 10 millimetres has radius 1 centimetre, area 100π square millimetres and area π square centimetres. There are 100 square millimetres in one square centimetre, even though there are only ten millimetres in one centimetre. Area conversions square the length factor. Similarly, doubling a radius doubles the circumference but multiplies the area by four. This scaling test is useful for checking a spreadsheet: if doubling radius merely doubles computed area, the formula is missing the square.
Distinguish a circle from an arc or segment
A circular arc is only part of the boundary, and a sector is the region between two radii and their arc. A circular segment is bounded by a chord and an arc. Radius alone does not tell you an arbitrary arc length or segment area; you also need an angle, chord or equivalent extra information. A turning circle for a vehicle is a practical clearance specification and may not equal the path of every point on the vehicle. Use the simple circle formulas only when the shape and the measurement definition match.
Use the calculator and verify the inverse
Enter the known radius in the Circle Calculator and check its displayed units. If your known value is diameter, circumference or area, first apply the inverse formula above. For the 5-centimetre example, divide the displayed circumference by 2π and confirm that it returns approximately five. Do the same with √(A/π) for area. Differences at the final decimal place can follow from output rounding. For a physical circle, measurement uncertainty may exceed that display precision, so record the original measurement rather than treating every displayed digit as equally reliable. For a measured diameter, check that the measuring line passes through the centre rather than using a shorter chord.
Use the Circle Calculator
Source and scope
Related guides
Use circle formulas in both directions, compare area and circumference, and add sector and arc relationships with an explicit angle convention.
Calculate a circular segment from radius and central angle, distinguish it from a sector, and use the correct radian formula with a worked example.
Calculate circular area directly from diameter, verify the radius step, and distinguish a disc from a ring or a noncircular shape with a diagonal.