Circle Calculator
Calculate area, circumference, diameter, radius, arc length, and sector area
Selected Mode
Calculate all circle properties from any known value Calculate arc length, sector area, chord length, and segment area
The distance from center to edge
The distance across the circle through center (d = 2r)
The distance around the circle (C = 2πr)
The space inside the circle (A = πr²)
Circle Properties
Radius (r)
Diameter (d)
Circumference (C)
Area (A)
Formulas Used
Diameter
d = 2r
Circumference
C = 2πr = πd
Area
A = πr²
Pi (π)
≈ 3.14159...
Arc & Sector Results
Arc Length
Sector Area
Chord Length
Segment Area
Angle Information
Angle in Degrees
Angle in Radians
Full Circle Comparison
Full Circumference
Full Area
Arc & Sector Formulas
Arc Length
L = rθ
Sector Area
A = ½r²θ
Chord Length
c = 2r·sin(θ/2)
Segment Area
A = ½r²(θ - sin θ)
Note: θ must be in radians for these formulas
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About Circle Calculator
What is a Circle?
A circle is a fundamental two-dimensional geometric shape consisting of all points in a plane that are at a constant distance (the radius) from a fixed central point. Circles appear everywhere in nature, architecture, engineering, and everyday life—from the wheels on your car to the orbits of planets.
How to Use This Calculator
Basic Properties Mode
- Select Basic Properties from the mode selector
- Choose which value you know: Radius, Diameter, Circumference, or Area
- Enter your known value in the input field
- The calculator instantly computes all other circle properties
Arc & Sector Mode
- Select Arc & Sector from the mode selector
- Choose your angle unit: Degrees or Radians
- Enter the Radius of your circle
- Enter the Central Angle of the arc or sector
- View calculated arc length, sector area, chord length, and segment area
Circle Formulas
Basic Circle Formulas
| Property | Formula | Description |
|---|---|---|
| Diameter | d = 2r | Diameter is twice the radius |
| Circumference | C = 2πr = πd | The perimeter or distance around the circle |
| Area | A = πr² = πd²/4 | The space enclosed within the circle |
Arc and Sector Formulas
| Property | Formula | Description |
|---|---|---|
| Arc Length | L = rθ | Length of the curved portion (θ in radians) |
| Sector Area | A = ½r²θ | Area of the "pie slice" region |
| Chord Length | c = 2r sin(θ/2) | Straight-line distance between arc endpoints |
| Segment Area | A = ½r²(θ - sin θ) | Area between chord and arc |
Understanding Circle Properties
Radius (r)
The radius is the distance from the center of a circle to any point on its edge. It's the most fundamental measurement of a circle, and all other properties can be derived from it.
Common uses:
- Determining the size of circular objects
- Calculating turning radius of vehicles
- Setting compass width for drawing circles
Diameter (d)
The diameter is the longest distance across a circle, passing through its center. It equals exactly twice the radius (d = 2r).
Common uses:
- Measuring pipe sizes in plumbing
- Specifying wheel sizes for vehicles
- Describing astronomical objects
Circumference (C)
The circumference is the total distance around the circle—its perimeter. The formula C = 2πr reveals the beautiful relationship between a circle's circumference and its radius, mediated by the constant π.
Common uses:
- Calculating the length of circular tracks
- Determining belt or band lengths
- Measuring waist sizes
Area (A)
The area represents the amount of space enclosed within the circle. The formula A = πr² shows that area grows with the square of the radius—double the radius, and the area quadruples.
Common uses:
- Calculating pizza sizes
- Determining material needed for circular objects
- Computing cross-sectional areas in engineering
The Mathematical Constant π (Pi)
Pi (π ≈ 3.14159265358979...) is perhaps the most famous mathematical constant. It represents the ratio of any circle's circumference to its diameter:
π = C ÷ d
Pi is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation never ends or repeats. Despite its infinite nature, we typically use approximations like 3.14, 22/7, or more precise values depending on the required accuracy.
Real-World Applications
Engineering & Construction
- Designing wheels, gears, and bearings
- Calculating pipe flow rates
- Planning circular structures and domes
Science & Nature
- Understanding planetary orbits
- Analyzing wave patterns
- Studying cell structures
Everyday Life
- Cooking and baking (pizza, cake sizes)
- Sports (running tracks, ball sizes)
- Art and design (circular logos, patterns)
Frequently Asked Questions
How do I find the radius from the area?
Rearrange the area formula A = πr² to solve for r:
r = √(A ÷ π)
For example, if the area is 100 square units, the radius is √(100 ÷ π) ≈ 5.64 units.
How do I find the radius from the circumference?
Rearrange the circumference formula C = 2πr to solve for r:
r = C ÷ (2π)
What's the difference between arc length and chord length?
- Arc length is the distance along the curved portion of the circle between two points
- Chord length is the straight-line distance between those same two points
The arc length is always greater than or equal to the chord length (they're equal only when the angle is 0).
What is a sector?
A sector is a "pie slice" shaped region of a circle, bounded by two radii and the arc between them. A pizza slice is a perfect example of a sector.
What is a segment?
A segment is the region between a chord and the arc it cuts off. It's the sector minus the triangle formed by the two radii and the chord.
How accurate is this calculator?
This calculator uses JavaScript's built-in Math library, which provides approximately 15-17 significant digits of precision for π and trigonometric functions. You can select 2, 4, 6, or 8 decimal places for displayed results.