Circle Calculator
Find radius, diameter, circumference, and area from any one of them, plus arc length, sector area, chords, central angles, and segments.
Radius, diameter, circumference, and area from any one of them.
From the center to the edge
| Radius (r) | 5 |
|---|---|
| Diameter (d) | 10 |
| Circumference (C) | 10π ≈ 31.416 |
| Area (A) | 25π ≈ 78.54 |
How this was calculated
Step 1 — Diameter
d = 2r
= 2 × 5 = 10
Step 2 — Circumference
C = 2πr
= 2 × π × 5
= 10π ≈ 31.416
Step 3 — Area
A = πr²
= π × 5²
= 25π ≈ 78.54
Arc, sector, or segment?
π ≈ 3.14159265359 is used at full double precision; values are rounded only for display, to 3 decimal places. For triangles inside a circle, try the Triangle Calculator; to convert units, use the Length Converter, Area Converter, or Angle Converter; and for spheres and cylinders, see the Surface Area Calculator.
This circle calculator works from whatever you know. Enter a radius, diameter, circumference, or area, and it finds the other three, showing exact answers in terms of π (such as 25π) alongside decimals. Advanced modes solve arc length and sector area from a radius and central angle, chord length from an angle or from the chord's distance to the center, the central angle from an arc, sector, or chord, and the area and height of a circular segment.
Every result comes with a labeled diagram and step-by-step working, in the unit of your choice.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Radius 5 cm | C = 10π ≈ 31.416 cm, A = 25π ≈ 78.54 cm² | d = 2 × 5 = 10 cm. C = 2π × 5 = 10π ≈ 31.416 cm. A = π × 5² = 25π ≈ 78.54 cm². |
| Circumference 10π | r = 5, d = 10 | r = C / (2π) = 10π / 2π = 5, so d = 10 and A = 25π ≈ 78.54. |
| Area 78.54 m² | r ≈ 5 m | r = √(78.54 / π) ≈ √25 ≈ 5.000 m, so d ≈ 10 m and C ≈ 31.42 m. |
| Quarter circle: r = 10, θ = 90° | Arc 5π ≈ 15.708, sector 25π ≈ 78.54 | L = (90/360) × 2π × 10 = 5π ≈ 15.708. Sector = (90/360) × π × 10² = 25π ≈ 78.54. Perimeter including the two radii: 5π + 20 ≈ 35.708. |
| Chord: r = 10, θ = 60° | Chord = 10 | c = 2 × 10 × sin(30°) = 20 × 0.5 = 10. With a 60° angle, the two radii and the chord form an equilateral triangle. |
| Chord 5 from the center of a radius-5 circle, distance 3 | Chord = 8 | c = 2√(5² − 3²) = 2√16 = 8. The distance, half the chord, and the radius form a 3-4-5 right triangle. |
| Segment: r = 10, θ = 90° | Area = 25π − 50 ≈ 28.54 | Sector 25π ≈ 78.54 minus the right triangle ½ × 10² × sin 90° = 50 gives ≈ 28.54. Height h = 10(1 − cos 45°) ≈ 2.93. |
Which Circle Formula Should I Use?
Radius: d = 2r, C = 2πr, A = πr².
Diameter: halve it for the radius (r = d / 2), then C = πd and A = πr².
Circumference: r = C / (2π), then the diameter and area.
Area: r = √(A / π), then the diameter and circumference.
Radius and central angle: arc length L = (θ/360) × 2πr, sector area (θ/360) × πr², and chord c = 2r sin(θ/2).
Radius and chord: the central angle θ = 2 arcsin(c / 2r).
Radius and the distance from the center to a chord: c = 2√(r² − d²).
Arc, Sector, Chord, and Segment
The radius runs from the center to the circle, and the diameter crosses the whole circle through the center. The circumference is the distance around the circle; for a circle this boundary length is what 'perimeter' means.
An arc is part of the circumference: a curved length. A chord is the straight line joining the two ends of an arc. A sector is the slice between two radii and an arc, like a slice of pizza. A segment is the region between a chord and its arc: the sector with the triangle between the two radii removed, so segment area = sector area − ½r² sin θ.
A common mistake is to use arc length as a perimeter. A semicircle of radius 10 has an arc length of 10π ≈ 31.42, but its perimeter, including the straight diameter, is 10π + 20 ≈ 51.42.
Circle Formula Reference
Diameter: d = 2r. Circumference: C = 2πr = πd. Area: A = πr².
Arc length: L = (θ/360) × 2πr for θ in degrees, or L = rθ for θ in radians.
Sector area: A = (θ/360) × πr² for degrees, or ½r²θ for radians.
Chord length: c = 2r sin(θ/2), or c = 2√(r² − d²) from the distance d to the center.
Sagitta (segment height): h = r(1 − cos(θ/2)).
Segment area: A = ½r²(θ − sin θ), with θ in radians. To convert, radians = degrees × π / 180.
Why Pi Appears in Every Circle Formula
π (pi) is the ratio of any circle's circumference to its diameter, about 3.14159265359. It is the same for every circle, which is why C = πd works for all of them, and it is irrational, so its decimals never end or repeat. That is why exact answers are often left in terms of π: a circle of radius 5 has area exactly 25π, which is about 78.54. This calculator uses π at full double precision and never asks you to enter it, though you can type it into any field.
Where Circle Calculations Are Used
Homework: find missing circle measurements and check each step. Construction: work out the circumference of a round column from its measured diameter, or the area of a circular patio. Manufacturing: estimate the surface area of discs and circular plates. Engineering: relate arcs, chords, and central angles on curved parts, or find how high an arch rises above its span (the sagitta). Design: size circular and pie-shaped elements for layouts and charts.
How to Use the Circle Calculator
- Choose Circle properties to work from one measurement, or pick Arc & sector, Chord, Central angle, or Segment for circle geometry problems.
- In Circle properties, choose what you know (radius, diameter, circumference, or area) and enter just that value. You can type π directly, for example 10π or 25pi.
- For angle-based modes, enter the radius and the central angle, and choose degrees or radians. Radians can be typed as multiples of π, such as π/2.
- Pick the unit of your measurements and, if you like, a different unit for the results. Areas are converted with the square of the length factor.
- Read the results with their exact π form where one exists, check the diagram, and follow the step-by-step working. Choose the number of decimal places to display.
Frequently Asked Questions
How do I calculate the area of a circle?
Square the radius and multiply by π: A = πr². For a radius of 5, the area is 25π ≈ 78.54. If you know the diameter, halve it first to get the radius.
How do I calculate the circumference of a circle?
Multiply the diameter by π, or twice the radius by π: C = πd = 2πr. A circle with radius 5 has circumference 10π ≈ 31.416.
How do I find the radius from the diameter?
Divide the diameter by 2: r = d / 2. A diameter of 10 cm gives a radius of 5 cm.
How do I find the radius from the circumference?
Divide the circumference by 2π: r = C / (2π). A circumference of 31.416 gives a radius of about 5. To find the radius from the area instead, use r = √(A / π).
What is the difference between radius and diameter?
The radius goes from the center to the edge; the diameter goes all the way across through the center, so it is always twice the radius.
What is an arc?
An arc is part of the circumference between two points on the circle. Its length is L = (θ/360) × 2πr, where θ is the central angle in degrees.
What is a sector?
A sector is the pie-slice region between two radii and the arc joining them. Its area is (θ/360) × πr²; a 90° sector is a quarter of the circle.
What is a chord?
A chord is a straight line joining two points on a circle. Its length is c = 2r sin(θ/2) from the central angle, or 2√(r² − d²) from its distance d to the center. The longest chord is the diameter.
How do I calculate arc length?
Multiply the circumference by the fraction of the circle the arc covers: L = (θ/360) × 2πr. With θ in radians it simplifies to L = rθ. For r = 10 and θ = 90°, L = 5π ≈ 15.708.
How do I calculate sector area?
Multiply the circle's area by the fraction of the circle: A = (θ/360) × πr². For r = 10 and θ = 90°, the sector area is 25π ≈ 78.54.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, so it reaches the center. A segment is bounded by a chord and an arc, so it does not; its area is the sector area minus the triangle formed by the two radii and the chord.
Why is pi used in circle calculations?
Pi is the ratio of every circle's circumference to its diameter, so it links a circle's straight measurements (radius and diameter) to its curved ones (circumference, arcs, and areas).
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Last updated: September 27, 2026.