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.
Circle values for radius 1 to 10
| Radius | Diameter | Circumference | Area |
|---|---|---|---|
| 1 | 2 | 6.28 | 3.14 |
| 2 | 4 | 12.57 | 12.57 |
| 3 | 6 | 18.85 | 28.27 |
| 4 | 8 | 25.13 | 50.27 |
| 5 | 10 | 31.42 | 78.54 |
| 6 | 12 | 37.70 | 113.10 |
| 7 | 14 | 43.98 | 153.94 |
| 8 | 16 | 50.27 | 201.06 |
| 9 | 18 | 56.55 | 254.47 |
| 10 | 20 | 62.83 | 314.16 |
Doubling the radius doubles the circumference but quadruples the area. A radius of 10 has an area of 314.16, four times the 78.54 of radius 5.
Finding one value from another
- Circumference from diameter: C = πd
- Diameter from circumference: d = C ÷ π
- Area from diameter: A = πd² ÷ 4
- Area from circumference only: A = C² ÷ 4π
- Circumference from area: C = 2√(πA)
Example: a circle with a circumference of 44 has a diameter of 14.01, a radius of 7.00 and an area of 154.06.
Square footage of a circle
Area is in square units, so convert every length to feet first. If you measure in inches, divide square inches by 144 to get square feet. One square metre is about 10.764 square feet.
| Item | Radius | Area |
|---|---|---|
| 10 ft diameter patio | r = 5 ft | π × 5² = 78.54 sq ft |
| 36 in round table | r = 18 in | π × 18² = 1,017.88 sq in ÷ 144 = 7.07 sq ft |
| 2 m diameter circle | r = 1 m | π × 1² = 3.14 m² × 10.764 = 33.82 sq ft |
| 20 ft diameter pool | r = 10 ft | π × 10² = 314.16 sq ft |
If you are buying flooring, paint or turf, many installers allow extra for cuts and waste, so check with your supplier.
Equation of a circle
A circle with centre (h, k) and radius r has the equation (x − h)² + (y − k)² = r². A circle of radius 5 at the origin is x² + y² = 25. A circle centred at (3, −2) with radius 4 is (x − 3)² + (y + 2)² = 16.
Does the value of π matter?
For a circle of radius 10:
| π used | Area | Circumference |
|---|---|---|
| 3.14 | 314.00 | 62.80 |
| 3.1416 | 314.16 | 62.832 |
| 3.14159265… | 314.16 | 62.832 |
Using 3.14 instead of the full value changes the area by about 0.05%. For homework use the value your teacher asks for. For building and design use the calculator's full precision.
Units
Lengths such as radius and circumference use the same unit as the input. Area uses that unit squared, for example cm² or ft². Do not mix units, such as a radius in feet and a height in inches.
Circle guides
- Area of a circle formula with examples
- How to find the square footage of a circle
- Circumference of a circle from diameter and radius
- Equation of a circle in standard form
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).
How do you find the area of a circle?
Use A = πr². A circle with radius 5 has an area of 78.54. If you know the diameter, use A = πd² ÷ 4.
How do you find the circumference from the diameter?
Multiply by π. A diameter of 10 gives a circumference of 31.42.
How do you find the diameter from the circumference?
Divide the circumference by π. A circumference of 44 gives a diameter of 14.01.
How do you find the area of a circle from its circumference only?
Use A = C² ÷ 4π. A circumference of 44 gives an area of 154.06.
How do I calculate the square feet of a circle?
Convert the radius to feet and use A = πr². A 10 ft diameter circle has a radius of 5 ft and an area of 78.54 sq ft. If you measure in inches, divide the area in square inches by 144.
What is the equation of a circle?
(x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. At the origin it simplifies to x² + y² = r².
Does using 3.14 for π change the answer much?
Only slightly. For radius 10 the area is 314.00 with 3.14 and 314.16 with the full value, a difference of about 0.05%.
Why does the area quadruple when the radius doubles?
Area depends on the radius squared. Radius 5 gives 78.54 and radius 10 gives 314.16, which is four times as much.
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