Circle Calculator

Enter any one known value – radius, diameter, circumference, or area – and instantly compute all other circle properties.Optionally provide a central angle (degrees/radians) to calculate arc length, sector area, chord length, and segment area, with real‑time visualisation.

Provide a central angle to compute arc length, sector area, and chord length. Toggle between degrees and radians.
⚪ Unit Circle (r=1) ? r=3.5 ? r=12 ? d=8 ? C=25 ? A=100 ? θ=45° r=6
Privacy first: All calculations are performed locally in your browser. No data is sent to any server.

The Circle: A Cornerstone of Geometry

The circle is one of the most fundamental and elegant shapes in mathematics – the set of all points in a plane that are equidistant from a fixed center point. This simple definition gives rise to a rich tapestry of properties, formulas, and applications that span nearly every field of science, engineering, and design. From ancient Greek geometry to modern computer graphics, the circle has always been a source of wonder and practical utility.

For a circle of radius r:

Diameter d = 2r  ·  Circumference C = 2πr  ·  Area A = πr²

Arc Length s = r·θ  ·  Sector Area Asector = ½ r² θ  ·  Chord Length c = 2r·sin(θ/2)  ·  Segment Area Asegment = ½ r² (θ − sin θ)

(θ in radians)

Mathematical Derivation: From Radius to Everything

The properties of a circle are elegantly interconnected. Starting from the radius r, the diameter is d = 2r; the circumference is C = 2πr, where π ≈ 3.141592653589793; and the area is A = πr², a result that can be proven by integration or by Archimedes' method of exhaustion.

When a central angle θ (in radians) is introduced, the arc length is s = r·θ, the sector area is Asector = ½ r² θ, the chord length is c = 2r·sin(θ/2), and the segment area is Asegment = ½ r² (θ − sin θ). These formulas are derived from basic trigonometry and are foundational in calculus, physics, and engineering.

Real‑World Applications

Mechanical Engineering – Gear Design

In gear design, the pitch circle defines the gear's size and determines speed ratios, torque, and centre distances. Engineers use circle calculations to quickly find pitch diameter, circumference (related to tooth spacing), and area (for material estimates). With a central angle, they can compute tooth arc lengths for optimised profiles. This calculator streamlines such iterative design tasks.

Everyday Life – Pizza Size Comparison

A 12‑inch pizza (diameter) has radius 6 inches and area ≈ 113.1 in²; a 16‑inch pizza has area ≈ 201.1 in² – that's about 78% more pizza for a modest price increase. The circle area formula helps consumers and businesses make informed decisions. Our calculator makes these comparisons instant and visual.

Common Misconceptions

  • Circumference vs. perimeter: For circles, “circumference” is the preferred term, but mathematically both mean the distance around the shape.
  • π equals exactly 3.14: π is irrational; 3.14 and 22/7 are approximations. This tool uses double‑precision floating‑point for high accuracy.
  • Diameter is twice the radius: Correct. However, radius is not half the circumference – that would be radius = C/(2π).
  • Sector vs. segment: The sector is the whole “pie slice” (two radii plus arc); the segment is the region between the chord and the arc. The segment area is always smaller than the sector area.

Rooted in classical and modern mathematics – This calculator follows the rigorous geometry of Euclid and Archimedes, and references authoritative sources such as Wolfram MathWorld and Wikipedia Circle. The interactive canvas uses standard HTML5 Canvas, and all computations are performed locally. Reviewed by the GetZenQuery tech team, July 2026.

References: MathWorld Circle; Wikipedia: Circle; Coxeter, H.S.M. Geometry Revisited (1967)