Circle Measurements Calculator
Enter any one circle measurement - radius, diameter, circumference, or area - and this calculator fills in all the others instantly. It also works out arc length and sector area for a given central angle, chord length and segment area, and the diameter of a circle with the same perimeter as a square. Switch between metric and imperial units at any time.
Formula
Worked example
A circle with radius 5 cm: diameter = 10 cm; circumference = 2 x pi x 5 = 31.416 cm; area = pi x 25 = 78.540 cm^2. For a 90-degree central angle: arc length = 5 x (pi/2) = 7.854 cm; sector area = 0.5 x 25 x (pi/2) = 19.635 cm^2; chord length = 2 x 5 x sin(45) = 7.071 cm.
What this calculator does
A circle is fully defined by a single measurement. Once you know the radius, diameter, circumference, or area, all the others follow from the same set of exact relationships. Enter whichever value you have in the "Known value" field, choose the right "Solve from" option, and the calculator displays all four main measurements plus the arc length, sector area, chord length, and segment area for the central angle you specify. Results update instantly as you type.
The four main circle measurements and how they relate
The radius (r) is the distance from the centre to any point on the circle. The diameter (d) is the longest possible chord and equals exactly 2r. The circumference (C) is the total distance around the circle and equals 2*pi*r, or equivalently pi*d. The area (A) is the surface enclosed inside the circle and equals pi*r^2. These four quantities are all direct functions of each other, so knowing one pins down the rest.
Arcs, sectors, chords and segments
A central angle (the angle formed at the centre of the circle by two radii) carves the circle into named regions. The arc is the curved portion of the circumference between the two radii; its length equals r times the angle in radians. The sector is the pie-slice shaped region between the two radii and the arc; its area equals half of r-squared times the angle in radians. The chord is the straight line connecting the two arc endpoints; its length equals 2r times the sine of half the central angle. The segment is the region between the chord and the arc; its area is the sector area minus the triangle area (0.5 * r^2 * sin(theta)).
Unit conversions and how to use them
All linear outputs (radius, diameter, circumference, arc, chord) are in the length unit you select. Area outputs are automatically in the square of that unit. For example, if you choose centimetres, radius is in cm and area is in cm^2. To convert between units, switch the "Length unit" dropdown and re-enter the same number, or convert the known value before entering it. Common conversion factors: 1 inch = 2.54 cm; 1 foot = 30.48 cm; 1 metre = 100 cm; 1 yard = 91.44 cm.
Practical applications
Circle measurement calculations appear in everyday tasks: finding the circumference of a wheel or tire (to calculate travel distance per rotation), working out how much material is needed to cover a circular surface (area), laying out a circular garden bed (radius from area or circumference), calculating the length of a curved track segment (arc length), sizing a circular table top (area or diameter), or determining the reach of a compass set to a known radius. The sector and segment formulas are commonly used in engineering when dealing with circular cuts, pipe cross-sections, and curved beam problems.
Circle formula reference
| Find | Given | Formula |
|---|---|---|
| Radius (r) | Diameter d | r = d / 2 |
| Radius (r) | Circumference C | r = C / (2π) |
| Radius (r) | Area A | r = √(A / π) |
| Diameter (d) | Radius r | d = 2r |
| Circumference (C) | Radius r | C = 2πr |
| Circumference (C) | Diameter d | C = πd |
| Area (A) | Radius r | A = πr² |
| Area (A) | Diameter d | A = πd² / 4 |
| Arc length (s) | r, angle θ (rad) | s = rθ |
| Sector area | r, angle θ (rad) | A_s = r²θ / 2 |
| Chord length | r, angle θ (rad) | l = 2r·sin(θ/2) |
Standard formulas relating the four main circle measurements. pi = 3.14159...
Frequently asked questions
How do I find the circumference if I only know the area?
First find the radius from the area: r = sqrt(A / pi). Then use C = 2 * pi * r. Or in one step, C = 2 * sqrt(pi * A). For example, if the area is 78.54 cm^2, then r = sqrt(78.54 / 3.14159) = 5.00 cm, and C = 2 * pi * 5 = 31.42 cm. Select "Area" in the "Solve from" dropdown and this calculator does that automatically.
What is the difference between arc length and circumference?
The circumference is the arc length of the complete circle (a 360-degree arc). An arc length is the circumference of a portion of the circle. For a central angle of theta degrees, the arc length is (theta / 360) times the full circumference, or equivalently r times the angle in radians.
How is a sector different from a segment?
A sector is the pie-slice region between two radii and the arc connecting them, like a slice of pie. A segment is the region between a chord and the arc, like the rounded cap you get when you cut across the circle with a straight line. The segment area equals the sector area minus the area of the triangle formed by the two radii and the chord.
How do I find the radius from the circumference?
Rearrange the circumference formula C = 2 * pi * r to get r = C / (2 * pi). For example, if a circle has a circumference of 62.83 cm, then r = 62.83 / (2 * 3.14159) = 10 cm.
Can I use this calculator for a semicircle?
Yes. Set the central angle to 180 degrees. The arc length will equal half the circumference, the sector area will equal half the full area, and the chord length will equal the diameter (the straight edge of the semicircle). The segment area at 180 degrees equals the sector area (which is half the circle) minus zero, since the triangle degenerates to a line.
What is the relationship between a circle and a square with the same perimeter?
A circle with the same perimeter (circumference) as a square has a larger area than the square. If a square has a side length s, its perimeter is 4s, and a circle with that circumference has radius r = 4s / (2 * pi) = 2s / pi. A circle always encloses more area for a given perimeter than any other shape - this is called the isoperimetric inequality.
How many decimal places should I use for pi?
For everyday calculations, pi = 3.14159 (5 decimal places) gives results accurate to better than 0.0001%. Engineering applications commonly use pi = 3.14159265 (8 decimal places). This calculator uses the full precision of pi available in JavaScript (about 15 significant digits), and you can set the display precision from 2 to 6 decimal places.