Area, circumference, radius and diameter of a circle from any one of them, plus arc length, chord, sector and segment area for a central angle.
By AdminDrafted with AI writing toolsUpdated Checked examples: 6
A circle of radius 5 cm has diameter 10 cm, circumference 31.4159 cm and area 78.5398 cm². A 60° arc of it is 5.236 cm long, and the chord joining its ends is 5 cm.
Aₛ is the sector area and Aɡ is the segment area. The segment is the region between the chord and the arc; θ is in radians inside the bracket.
Give any one of the radius, diameter, circumference or area and the other three follow from d = 2r, C = 2πr and A = πr². For a central angle θ the calculator also returns the arc length s = rθ (θ in radians), the chord c = 2r sin(θ/2), the sector area πr² × θ/360° and the segment area r²/2 × (θ − sin θ).
The default radius of 5 cm gives a diameter of 10 cm, a circumference of 31.4159 cm and an area of 78.5398 cm². Its 60° arc is 5.236 cm long and the chord across it is exactly 5 cm, because two radii and that chord form an equilateral triangle. Starting from the circumference suits trees, pipes and columns whose diameter cannot be reached: a girth of 100 cm means a diameter of 31.831 cm.
Inputs can be in any length or area unit, including hectares and acres, and every result appears in the unit chosen under "Show results in". The central angle must be above 0° and at most 360°; at 360° the segment is the whole disc.
The area of a circle is A = πr² and its circumference is C = 2πr, or πd using the diameter. A circle with a 3 cm radius covers π × 3² ≈ 28.27 cm² and measures 2π × 3 ≈ 18.85 cm around. Given a diameter, halve it before squaring.
π is the ratio of any circle's circumference to its diameter, about 3.14159265 (NIST DLMF §3.12).
| Radius, cm | Diameter, cm | Circumference, cm | Area, cm² |
|---|---|---|---|
| 1 | 2 | 6.283185 | 3.141593 |
| 2 | 4 | 12.566371 | 12.566371 |
| 3 | 6 | 18.849556 | 28.274334 |
| 5 | 10 | 31.415927 | 78.539816 |
| 6 | 12 | 37.699112 | 113.097336 |
| 10 | 20 | 62.831853 | 314.159265 |
| 100 | 200 | 628.318531 | 31415.926536 |
Doubling the radius doubles the circumference but multiplies the area by four: a 10 cm radius encloses 314.159265 cm², four times the 5 cm circle. At r = 2 both columns read 4π ≈ 12.566371, one in cm and the other in cm².
Divide the circumference by 2π. A tree trunk 150 cm around has r = 150/(2π) ≈ 23.87 cm and a diameter of about 47.75 cm.
Divide the area by π and take the square root. A circle covering one hectare, 10,000 m², has r = √(10,000/π) ≈ 56.42 m; one acre gives about 35.89 m.
For a central angle θ in degrees, the arc length is s = 2πr × θ/360°, the sector area is πr² × θ/360° and the chord is c = 2r sin(θ/2). In radians the first two become s = rθ and r²θ/2.
Every row below uses a 5 cm radius, with lengths in centimeters and areas in square centimeters.
| Angle | Arc, cm | Chord, cm | Sector, cm² | Segment, cm² |
|---|---|---|---|---|
| 30° | 2.617994 | 2.58819 | 6.544985 | 0.294985 |
| 60° | 5.235988 | 5 | 13.089969 | 2.264652 |
| 90° | 7.853982 | 7.071068 | 19.634954 | 7.134954 |
| 120° | 10.471976 | 8.660254 | 26.179939 | 15.354621 |
| 180° | 15.707963 | 10 | 39.269908 | 39.269908 |
| 270° | 23.561945 | 7.071068 | 58.904862 | 71.404862 |
| 360° | 31.415927 | 0 | 78.539816 | 78.539816 |
A 60° chord equals the radius, because the two radii and the chord form an equilateral triangle. At 180° the chord is the diameter, and at 360° it shrinks to zero while the arc becomes the whole circumference.
A sector is the pie slice between two radii and the arc. A segment is the region between the arc and its chord: the sector minus the triangle formed by the two radii, with area r²(θ − sin θ)/2 for θ in radians (MathWorld).
At 90° on the 5 cm circle, the triangle covers ½ × 5 × 5 = 12.5 cm², so the segment is 19.634954 − 12.5 = 7.134954 cm². Past 180° the segment is the larger piece, which is why at 270° it exceeds the sector.
For other flat shapes, use the area and perimeter calculator. To find a circle from three points on its edge, use the circle-through-three-points calculator.
Checked against: Independent calculation
Checked against: Independent calculation
Checked against: sqrt(1/pi), 2*sqrt(pi)
Checked against: 12*sin(pi/4), 36*pi/4, 36*pi/4 − 18, 6*pi/2
Use A = πd²/4, which is πr² with the radius written as half the diameter. A 10 cm diameter gives 25π ≈ 78.54 cm², and a 12 in pizza covers 36π ≈ 113.10 in². Area grows with the square of the diameter, so one 16 in pizza (201.06 in²) has more area than two 11 in pizzas together (190.07 in²).
Divide the circumference by 2π: r = C/(2π). A tree with a girth of 100 cm has a radius of 15.915 cm and a diameter of 31.831 cm (C/π). Foresters' diameter tapes do this division for you: their scale is graduated in units of π, so wrapping the tape round the trunk reads the diameter directly.
A sector is the pie slice between two radii and the arc; a segment is the region between the arc and the chord joining its ends. The segment is the sector minus the triangle formed by the two radii and the chord. For a 90° slice of a circle with a 6 in radius, the sector is 9π ≈ 28.274 in², the triangle 18 in², and the segment 10.274 in².
Multiply the radius by the central angle in radians: s = rθ. With the angle in degrees, use s = 2πr × θ/360. A 60° arc on a 5 cm radius is 5 × π/3 ≈ 5.236 cm, one sixth of the 31.416 cm circumference. Putting degrees straight into s = rθ gives an answer 57.3 times too large, the number of degrees in one radian.
For estimates, yes. 3.14 is 0.05% below π, so a 5 cm radius gives an area of 78.5 cm² instead of 78.54 cm²; the fraction 22/7 is 0.04% too high. The absolute error grows with size: for a radius of 100 m, 3.14 gives 31,400 m², which is 15.9 m² short of the true 31,415.9 m².
This calculator includes 6 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.