Circle Calculator
Enter any one of a circle's radius, diameter, circumference or area and get the other three — exact in terms of π, with the decimal and every step. Then add a central angle to find the arc length, sector area, chord, circular segment and segment height, with a labeled figure. Free, no sign-up; it runs in your browser.
Whole numbers, decimals and fractions such as 3/4. Circumference and area may be written as a multiple of π, such as 10π, to keep the answer exact.
Results
| Measurement | Exact | Decimal |
|---|---|---|
| Radius (r) | 5 cm | 5 cm |
| Diameter (d) | 10 cm | 10 cm |
| Circumference (C) | 10π cm | 31.42 cm |
| Area (A) | 25π cm² | 78.54 cm² |
The working
d = 2r = 2 × 5 = 10
C = 2πr = 2 × π × 5 = 10π
A = πr² = π × 5² = 25π
Arc, sector, chord and segment
These use the radius found above.
| Measurement | Exact | Decimal |
|---|---|---|
| Arc length (s) | 5π/3 cm | 5.24 cm |
| Sector area | 25π/6 cm² | 13.09 cm² |
| Chord length (c) | 5 cm | 5 cm |
| Segment area | 25π/6 − 25√3/4 cm² | 2.26 cm² |
| Segment height (sagitta) | 5 − 5√3/2 cm | 0.67 cm |
| Sector perimeter | 10 + 5π/3 cm | 15.24 cm |
| Segment perimeter | 5 + 5π/3 cm | 10.24 cm |
Working for the angle
θ = 60° = π/3 rad
s = r × θ = 5 × (π/3) = 5π/3
Sector = ½ × r² × θ = ½ × 5² × (π/3) = 25π/6
Chord = 2 × r × sin(θ / 2) = 2 × 5 × sin(30°) = 2 × 5 × 1/2 = 5
Segment = ½ × r² × (θ − sin θ) = ½ × 5² × (π/3 − √3/2) = 25π/6 − 25√3/4
Sagitta = r × (1 − cos(θ / 2)) = 5 − 5√3/2
Figure
How to Use the Circle Calculator
- 1. Say what you know. Choose radius, diameter, circumference or area, and type the value. Whole numbers, decimals and fractions such as 3/4 are read exactly.
- 2. Keep it exact with π. A circumference or area can be typed as a multiple of π, such as 10π or 25π, and the radius comes out as a clean number. Typed as a plain number, a circumference of 10 gives r = 5/π.
- 3. Read the four measurements. Radius, diameter, circumference and area appear in exact form and as decimals rounded to the places you choose, with the working beneath.
- 4. Add a central angle. Type it in degrees, or in radians as a multiple of π such as pi/3. The arc, sector, chord, segment and sagitta follow, drawn on a labeled circle.
Circle Formulas
| Find | Formula | Notes |
|---|---|---|
| Diameter | d = 2r | Twice the radius |
| Circumference | C = 2πr = πd | The distance around |
| Area | A = πr² | The surface inside |
| Radius from circumference | r = C / (2π) | Rearranges C = 2πr |
| Radius from area | r = √(A / π) | Rearranges A = πr² |
| Area from circumference | A = C² / (4π) | Substitutes r = C / (2π) |
The number π is the ratio of any circle's circumference to its diameter, about 3.14159. It is irrational, so the exact answer to most circle problems is written with π in it — 25π rather than 78.5398… — and rounded only at the end.
Arc, Sector, Chord and Segment
A central angle θ cuts the circle into pieces. Measure θ in radians (180° = π radians) for the cleanest formulas.
| Piece | Formula | What it is |
|---|---|---|
| Arc length | s = r × θ | The curved edge between the two radii |
| Sector area | ½ × r² × θ | The slice from the center to the arc |
| Chord | c = 2r × sin(θ / 2) | The straight line joining the arc's ends |
| Segment area | ½ × r² × (θ − sin θ) | The region between the chord and the arc |
| Sagitta | h = r × (1 − cos(θ / 2)) | The height of the segment |
The segment is the sector minus the triangle formed by the two radii and the chord, whose area is ½r² sin θ — which is where θ − sin θ comes from. For θ above 180° the formulas still hold: the "segment" is then the larger piece and its sagitta exceeds the radius.
Worked Examples
- From the radius. r = 5 gives d = 10, C = 10π ≈ 31.42 and A = 25π ≈ 78.54.
- From a circumference. C = 10 gives r = 10 / (2π) = 5/π ≈ 1.59, and A = C² / (4π) = 25/π ≈ 7.96. Typed as 10π, the same circle has r = 5.
- A 60° sector of radius 3. θ = π/3, so the arc is π ≈ 3.14, the sector is 3π/2 ≈ 4.71 and the chord is 2 × 3 × sin 30° = 3 — equal to the radius, because the triangle is equilateral.
- Its segment. Segment area = ½ × 9 × (π/3 − √3/2) = 3π/2 − 9√3/4 ≈ 0.82, and the sagitta is 3 − 3√3/2 ≈ 0.40.
Exact chords and segments appear when sin(θ/2) and sin θ have a closed form — that is, for angles that are multiples of 30° and 15° respectively. For other angles, such as 50°, the arc and sector are still exact but the chord and segment are given as decimals.
Common Mistakes
- Using the diameter in a formula that needs the radius: A = πr², not πd².
- Calculating with degrees in s = rθ or ½r²θ, which need radians; the calculator converts for you.
- Confusing the sector (a slice reaching the center) with the segment (the cap cut off by a chord).
- Rounding π to 3.14 early: the exact answer 25π and the decimal 78.54 differ by about 0.04.
Frequently Asked Questions
- How do you find the area of a circle from its circumference?
- First find the radius, r = C / (2π), then use A = πr². Combining them gives A = C² / (4π). A circumference of 10 gives an area of 100 / (4π) = 25/π, about 7.96.
- What is the difference between radius and diameter?
- The radius runs from the center to the edge; the diameter runs all the way across through the center, so d = 2r. Area formulas use the radius, while C = πd uses the diameter.
- How do you find the arc length of a circle?
- Multiply the radius by the central angle in radians: s = rθ. If the angle is in degrees, take the fraction of a full turn: s = (θ / 360°) × 2πr. A 60° arc of a circle with radius 3 is π long.
- What is the difference between a sector and a segment?
- A sector is the slice bounded by two radii and the arc, like a slice of pizza. A segment is the region between a chord and the arc it cuts off. The segment is the sector minus the triangle formed by the two radii and the chord.
- How do you find the length of a chord?
- c = 2r × sin(θ / 2), where θ is the central angle. If you know the chord's distance d from the center instead, c = 2√(r² − d²), by the Pythagorean theorem. A chord of 180° is a diameter.
- What is the sagitta of a circular segment?
- The sagitta is the height of the segment: the distance from the midpoint of the chord to the midpoint of the arc, h = r × (1 − cos(θ / 2)). It is the "bulge" of an arch or a lens.
- Why does the calculator show π instead of a decimal?
- Because π is irrational, any decimal is an approximation, and it also hides the structure of the answer. Writing 25π keeps the result exact for further work; the decimal is shown beside it for when you need a number to measure with.