Calculate radius, diameter, circumference, area, arc length, sector area, and chord length from any known circle measurement. Supports US customary and UK metric units.
The circle is the most fundamental shape in mathematics and appears everywhere — from coins to wheels, pipes to pools, clock faces to planetary orbits. Our free circle calculator computes every important circle measurement from just one known value: radius, diameter, circumference, or area. It also calculates arc length, sector area, and chord length for any central angle. Whether you are working in US customary units (inches, feet, miles) or UK/metric units (cm, m, km), the calculator adapts instantly.
All circle calculations derive from one fundamental relationship: the ratio of a circle's circumference to its diameter always equals π (pi ≈ 3.14159265...).
| Property | Formula | Where |
|---|---|---|
| Diameter | d = 2r | r = radius |
| Circumference | C = 2πr = πd | π ≈ 3.14159 |
| Area | A = πr² | r = radius |
| Arc length | s = rθ | θ in radians; s = (θ/360)×2πr for degrees |
| Sector area | K = ½r²θ | θ in radians; K = (θ/360)×πr² for degrees |
| Chord length | c = 2r·sin(θ/2) | θ = central angle in radians |
Pi (π) is an irrational number — it cannot be expressed as a simple fraction and its decimal expansion never repeats or terminates. The value begins 3.14159265358979... and continues infinitely. Mathematicians have calculated π to trillions of decimal places, yet for almost all practical engineering and science purposes, 3.14159 or even 22/7 (≈ 3.14286) is sufficient.
The symbol π was popularised by the Welsh mathematician William Jones in 1706 and adopted by Leonhard Euler, whose influence made it universal. Ancient Egyptians and Babylonians used approximations of π thousands of years ago — the Rhind Papyrus (c. 1650 BC) uses a value of 256/81 ≈ 3.16.
The unit circle is a circle with radius exactly 1, centred at the origin (0,0) in a coordinate system. It is the foundation of trigonometry: for any angle θ, the point where the terminal ray meets the unit circle has coordinates (cos θ, sin θ). This connects circle geometry to the six trigonometric functions and makes the unit circle essential in A-level Maths (UK), US pre-calculus, physics, and engineering.
The equation of a unit circle is: x² + y² = 1. More generally, a circle with centre (h, k) and radius r has equation: (x − h)² + (y − k)² = r².
A sector is a "pie slice" of a circle defined by two radii and the arc between them. The central angle θ determines what fraction of the full circle the sector represents.
Example: For a circle with radius 10 cm and central angle 90°: arc length = (90/360) × 2π×10 = ½π×10 ≈ 15.71 cm; sector area = (90/360) × π×100 = 25π ≈ 78.54 cm²; chord length = 2×10×sin(45°) ≈ 14.14 cm.
An annulus is the region between two concentric circles with different radii R (outer) and r (inner). It is the shape of a washer, a drainpipe cross-section, or a circular border.
Annulus area = π(R² − r²)
Annulus circumference = 2πR (outer) + 2πr (inner)
A car tyre with diameter 650 mm has circumference C = π × 650 ≈ 2,042 mm = 2.04 m. So at 60 mph (96.6 km/h), the tyre rotates approximately 96,600 / 2.04 ≈ 47,353 times per hour, or about 789 times per minute. Understanding tyre circumference is important for speedometer calibration. In the UK, tyre sizes follow ETRTO standards; in the US, P-metric sizing is standard — but the fundamental circumference formula is identical.
Plumbers and engineers calculate pipe cross-sectional area using A = πr² to determine flow rate capacity. A 4-inch (10.16 cm) diameter pipe has radius 2 inches (5.08 cm) and cross-sectional area ≈ 12.57 sq in (81.07 cm²). UK plumbing uses nominal bore sizes in millimetres; US plumbing uses nominal sizes in inches — but again the circle formula applies equally.
A circular swimming pool with radius 4 m has area = π × 16 ≈ 50.27 m² and circumference ≈ 25.13 m. Knowing the area allows calculation of water volume (area × depth) and the circumference gives the length of fencing or pool surround needed. In the UK, pool areas are measured in square metres; in the US, square feet are standard (1 m² = 10.764 ft²).
| Measurement | US Imperial | UK/Metric |
|---|---|---|
| Radius of 5 ft circle | r = 5 ft; C = 31.42 ft; A = 78.54 ft² | r = 1.524 m; C = 9.58 m; A = 7.30 m² |
| Radius of 10 in circle | r = 10 in; C = 62.83 in; A = 314.16 in² | r = 25.4 cm; C = 159.6 cm; A = 2026.8 cm² |
| Radius of 1 m circle | r = 39.37 in; C = 247.3 in; A = 1550 in² | r = 1 m; C = 6.283 m; A = 3.142 m² |
In physics, objects moving in circles experience centripetal acceleration directed toward the centre. The circumference formula C = 2πr directly gives the distance travelled in one revolution. For a satellite orbiting at radius r from Earth's centre at velocity v, its orbital period T = 2πr/v. The same principle applies to fairground rides, centrifuges, and planetary orbits — understanding the relationship between radius, circumference, and angular velocity is fundamental to classical mechanics.
In the Cartesian coordinate system, a circle with centre (h, k) and radius r is described by: (x − h)² + (y − k)² = r². Expanding this gives the general form: x² + y² + Dx + Ey + F = 0, where the centre is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F). This is a standard topic in UK A-level Pure Mathematics and US Pre-Calculus/Algebra 2.
Area = πr², where r is the radius. If you know the diameter d, use r = d/2 first. If you know the circumference C, use r = C/(2π). Example: a circle with radius 7 cm has area = π × 49 ≈ 153.94 cm².
Circumference C = 2πr = πd, where r is the radius and d is the diameter. Using π ≈ 3.14159: a circle with diameter 10 inches has circumference ≈ 31.42 inches.
Rearrange C = 2πr to get r = C/(2π). For a circumference of 50 cm: r = 50/(2π) ≈ 7.958 cm. Similarly, from area A = πr²: r = √(A/π).
Arc length is the distance along the curved part of a circle between two points. Arc length = (central angle / 360°) × 2πr for degree inputs, or arc length = r × θ for radian inputs. A 90° arc on a circle of radius 5 has length = (90/360) × 2π×5 = π×2.5 ≈ 7.854 units.
A sector is the "pie slice" region bounded by two radii and the arc between them. Sector area = (θ/360°) × πr² for degrees, or ½r²θ for radians. The perimeter of a sector = 2r + arc length.
An arc is the curved part of the circle between two points. A chord is the straight line connecting those same two points. Chord length = 2r·sin(θ/2) where θ is the central angle. The arc is always longer than (or equal to) the chord for the same two points.
Pi (π ≈ 3.14159) appears in every circle formula because it is defined as the ratio of a circle's circumference to its diameter. Every formula — C = 2πr, A = πr², arc = rθ, sector = ½r²θ — uses π because circles are fundamentally defined by this constant ratio.
An annulus is the ring-shaped region between two concentric circles. If the outer radius is R and inner radius is r, then the annulus area = π(R² − r²) and the total perimeter = 2πR + 2πr. Annuli appear in washers, pipe cross-sections, ring-shaped garden beds, and circular borders.
Results are based on standard mathematical formulas and the inputs you provide. Always verify measurements independently for engineering, construction, or safety-critical applications.