Circle Calculator
Enter any one of radius, diameter, circumference or area to get the other three, or work out a sector (arc length and area), a segment from its chord and height, or a ring (annulus) — with exact values in terms of π and every step shown.
Written by Suhaib Hassan. All formulas are standard Euclidean geometry, and each worked example is re-derived by hand. Our methodology.
It converts your input to the radius, then applies C = 2πr and A = πr², with sector, chord and segment formulas from the central angle.
One page covers the whole circle family — full circle, sector, segment and ring — and draws each to scale so you can check the geometry.
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Diagram (drawn to scale)
The circle with its radius and diameter marked.
Step-by-step working
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Circle formulas at a glance
Perimeter, area and angle range for each shape in the circle family (θ in radians).
| Shape | Boundary / arc | Area | Angle |
|---|---|---|---|
| Full circle | C = 2πr = πd | A = πr² | θ = 2π (360°) |
| Semicircle | πr + 2r | ½πr² | θ = π (180°) |
| Quadrant | ½πr + 2r | ¼πr² | θ = π/2 (90°) |
| Circular sector | s = rθ (arc length) | A = ½r²θ | 0 < θ ≤ 2π |
| Circular segment | chord c = 2r·sin(θ/2) | A = ½r²(θ − sin θ) | 0 < θ ≤ 2π |
| Annulus | 2π(R + r) | A = π(R² − r²) | R > r > 0 |
Worked examples
Bicycle wheel
A wheel has a 26-inch diameter. How far does it roll in one turn?
Sector of a pizza
A pizza of radius 10 in is cut into quarters (θ = 90°).
sector area = ½r²θ = 50 × π/2 = 78.540 in²
Bolt circle spacing
Eight equally spaced bolts on a 100 mm pitch-circle diameter (R = 50 mm).
= 100 × sin 22.5° = 38.27 mm
For a chord of 10 and a height of 2: r = 100 ÷ 16 + 1 = 7.25. Use the Segment tab to see the rest.
What is π and why does it appear?
π (pi) is the ratio of any circle’s circumference to its diameter, C ÷ d ≈ 3.14159. Because all circles are scaled copies of one another, that ratio is the same for every circle. Johann Lambert proved in 1761 that π is irrational, and Ferdinand von Lindemann proved in 1882 that it is transcendental.
Key formulas
- Diameter and radius: d = 2r.
- Circumference: C = 2πr = πd.
- Area: A = πr² = C² ÷ 4π.
- Sector: arc length s = rθ and area ½r²θ, with θ in radians (radians = degrees × π ÷ 180).
- Segment: area ½r²(θ − sin θ); chord c = 2r sin(θ/2); height h = r(1 − cos(θ/2)).
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Archimedes, Measurement of a Circle (c. 250 BCE), Proposition 1 — the area of a circle equals that of a right triangle with legs r and C.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI) — the radian as a unit of plane angle.
Frequently asked questions
Why is circumference divided by diameter always π?
Every circle is a scaled copy of every other. Scaling multiplies the circumference and the diameter by the same factor, so their ratio never changes. That constant ratio is π.
How do I find the radius from the area or circumference?
From the area, r = √(A ÷ π). From the circumference, r = C ÷ 2π. This calculator does both: choose what you know in the first tab.
What is the difference between a sector and a segment?
A sector is a “pie slice” bounded by two radii and an arc. A segment is the region between an arc and its chord. The segment area is the sector area minus the triangle formed by the two radii and the chord.
How do I find a segment’s area without knowing the angle?
From the chord c and height h, first find r = c² ÷ (8h) + h ÷ 2, then the central angle θ = 2·atan2(c ÷ 2, r − h), and use the segment formula ½r²(θ − sin θ). The Segment tab does this for you.
Should I use degrees or radians?
The arc-length and sector-area formulas s = rθ and ½r²θ require radians. If your angle is in degrees, multiply by π ÷ 180 first — or just pick “deg” on the Sector tab and the calculator converts for you.
What is an annulus?
An annulus is the ring between two concentric circles of radii R (outer) and r (inner). Its area is π(R² − r²), which can also be written π(R − r)(R + r).
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