Circle Calculator
Calculate area, circumference, diameter, radius, arc length, sector area, and chord length from any known value.
About the Calculator
This page is designed to cover the full circle cluster in one place: basic circle values, arc and sector geometry, equation conversion, and a practical unit-circle reference. Enter one known value to derive the rest instantly, then use the worked formulas and examples to understand the math behind each output. Use the Circle Calculator to get a clear result you can act on right away. This calculator is designed to be practical, fast, and easy to use on any device. If you are comparing options, run a few scenarios to see how small changes affect the outcome.
How to use this circle calculator
- 1Choose the tab that matches what you know: Basic Circle, Arc and Sector, Equation, or Unit Circle.
- 2Enter one known value in Basic Circle to auto-derive radius, diameter, circumference, and area.
- 3For Arc and Sector, enter radius and central angle, then pick degrees or radians.
- 4Use Equation mode to convert between standard and general forms and identify center/radius.
- 5Use Unit Circle mode to review exact trig values at standard angles for quick homework checks.
The Formula
| From | To | Formula |
|---|---|---|
| Radius (r) | Diameter (d) | d = 2r |
| Radius (r) | Circumference (C) | C = 2πr |
| Radius (r) | Area (A) | A = πr² |
| Diameter (d) | Radius (r) | r = d/2 |
| Diameter (d) | Circumference (C) | C = πd |
| Diameter (d) | Area (A) | A = π(d/2)² |
| Circumference (C) | Radius (r) | r = C/(2π) |
| Circumference (C) | Area (A) | A = C²/(4π) |
| Area (A) | Radius (r) | r = √(A/π) |
| Area (A) | Diameter (d) | d = 2√(A/π) |
| Arc inputs (r, θ) | Arc length (L) | L = rθ (rad) or (θ/360)·2πr (deg) |
| Arc inputs (r, θ) | Sector area | A_sector = ½r²θ (rad) or (θ/360)·πr² (deg) |
| Arc inputs (r, θ) | Chord length | ch = 2r·sin(θ/2) |
| Arc inputs (r, θ) | Segment area | A_seg = ½r²(θ − sinθ) |
| Center (h, k), radius r | Standard form | (x − h)² + (y − k)² = r² |
| Coefficients D, E, F | General form | x² + y² + Dx + Ey + F = 0 |
What is a circle?
A circle is the set of all points in a plane that are exactly the same distance from one fixed point, called the center. That constant distance is the radius. From that one definition, all core circle relationships follow: the diameter is twice the radius, circumference measures distance around the boundary, and area measures the enclosed surface.
Parts of a circle
Center: fixed point equidistant from all points on the circle.
Radius: segment from center to circle boundary.
Diameter: segment through center with both endpoints on circle.
Chord: any segment with both endpoints on the circle.
Arc: curved part of the circumference between two points.
Sector: pie-slice region bounded by two radii and an arc.
Segment: region between a chord and its corresponding arc.
Tangent/Secant: tangent touches once; secant cuts through two points.
The unit circle
The unit circle has radius 1 and center at (0,0). Any point on it maps angle θ to coordinates (cosθ, sinθ), which is why it is central to trigonometry. The unit-circle tab surfaces exact values at the standard angles so you can compare symbolic values and decimals quickly.
Equation of a circle
Standard form: (x - h)² + (y - k)² = r² - directly exposes the center (h, k) and radius r.
General form: x² + y² + Dx + Ey + F = 0 - convenient for polynomial work. The two forms are equivalent; convert by expanding standard form or completing the square on the general form.
Examples
Pizza slice sector area
A 14-inch pizza has radius 7 inches. One of 8 equal slices is a 45° sector. Sector area = (45/360) × π × 7² ≈ 19.24 square inches.
Running track curve length
A semicircular end with radius 36.5 m has arc length L = πr ≈ 114.67 m. Two semicircles form one full circle of circumference 2πr.
Irrigation field coverage
A center-pivot arm of 400 m is the radius. Area covered = π × 400² ≈ 502,655 square meters, or about 50.27 hectares.
Bike wheel travel per rotation
A wheel with 0.35 m radius travels one circumference each turn: C = 2π(0.35) ≈ 2.199 m per full rotation.
FAQ
What is the area of a circle?
The area is A = πr², where r is radius. You can also write A = πd²/4 when diameter is known.
How do I find circumference of a circle?
Use C = 2πr if radius is known, or C = πd if diameter is known. Both formulas are equivalent.
What is the difference between radius and diameter?
Radius goes from center to boundary. Diameter goes boundary-to-boundary through the center and equals 2r.
What is the formula for circle area?
The primary formula is A = πr². Alternate forms include A = πd²/4 and A = C²/(4π).
What is the circle equation?
Standard form is (x - h)² + (y - k)² = r². General form is x² + y² + Dx + Ey + F = 0.
What is the unit circle?
It is the circle of radius 1 centered at the origin. Coordinates on it define cos and sin values for angles.
How do I calculate arc length?
Arc length is L = rθ when θ is in radians. In degrees, L = (θ/360)·2πr.
What is a sector of a circle?
A sector is the region enclosed by two radii and the arc between them, like a pie slice.
What is pi and why is it used in circle formulas?
Pi (π) is the constant ratio of circumference to diameter for every circle, which is why it appears in all circle equations.
What is a circle graph?
A circle graph is another term for a pie chart. It represents categories as sectors proportional to each category value.
💡 Tips
- •Keep units consistent throughout each problem. Mixed units are the most common source of avoidable mistakes.
- •If an angle is in degrees, convert to radians before using L = rθ or A = ½r²θ.
- •When area is given, solve radius first with r = √(A/π), then derive all other values from r.
- •For equation problems, standard form is easier for reading center and radius; general form is easier for polynomial workflows.
- •Use exact forms (like π/6 or √3/2) for classroom work, then decimals for measurements and engineering estimates.
- •For quick checks: diameter should always equal twice radius; circumference should always be a little over 3 times diameter.
Pi - the most famous number in math
- •Pi is irrational, meaning it cannot be written as a finite or repeating decimal.
- •Common approximations include 3.14, 22/7, and 355/113.
- •Pi Day is celebrated on March 14 (3/14).
- •Using more pi precision improves numeric outputs, but for most practical measurements 3.14 is enough.
Related Calculators
- •Right Triangle Calculator - Pythagorean theorem, angles, and area.
- •Fraction Calculator - simplify and combine fractional values used in geometry.
- •Square Root Calculator - useful for reverse-circle formulas.
- •Percentage Calculator - convert sector fractions and percentages.
- •Average Calculator - summarize repeated measurements.
- •GPA Calculator - another math workflow for students using this page.