Circle Calculator – Calculate Radius, Diameter, Circumference, Area, Arc & Sector

  • Adam
  • November 4, 2025

Free online circle calculator. Calculate radius, diameter, circumference, area, arc length, and sector area. Universal calculator accepts any one value and computes all circle properties.

Circle Calculator

Calculate All Circle Properties: Radius, Diameter, Circumference, Area, and Circle Parts

Quick Navigation

Circle Basic Properties Calculator

Enter any one value (radius, diameter, or circumference) to calculate all circle properties.

Universal Circle Calculator

Enter only ONE value. Leave others blank.

Radius (R)
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Diameter (D)
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Circumference (C)
-
Area (A)
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Visual Reference
R D C
Key Relationships:
D = 2R
C = 2πR
A = πR²

Circle Area Calculator

Calculate circle area from radius or diameter.

Area Calculator
OR
Area (A)
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Area Breakdown
r
Area Formula:
A = πr²

Example:
If r = 5, then
A = π × 5² = 78.54

Arc & Sector Calculator

Calculate arc length and sector area from radius and angle.

Arc & Sector Calculator
Arc Length
-
Sector Area
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Sector Visualization
θ r
Arc & Sector Formulas:
Arc = (θ/360°) × 2πr
Sector = (θ/360°) × πr²

Understanding Circles

A circle is a simple closed shape where all points on its perimeter are equidistant from a central point. This perfect symmetry makes circles one of the most fundamental shapes in mathematics, appearing throughout geometry, physics, engineering, and nature.

Circles have unique properties that distinguish them from other shapes. They have no edges or corners, only a continuous curve. The center is equidistant from every point on the circle. This symmetry defines all circle calculations—everything depends on the radius, the distance from center to edge.

The relationship between circumference (distance around) and diameter (distance across) is constant: the ratio equals π (pi), approximately 3.14159. This mathematical constant appears in every circle formula, connecting radius to circumference and area.

Key Principle: All circle properties (radius, diameter, circumference, area) relate through the mathematical constant π.

Key Features & Capabilities

Universal Calculator Enter any value (R, D, C) and calculate all circle properties
Area Computation Calculate area from radius or diameter
Arc & Sector Find arc length and sector area with angle support
Degree & Radian Support for both angle measurement units
Step-by-Step Detailed calculations shown for every result
Visual Guides Diagrams showing circle parts and relationships

How to Use This Calculator

Basic Properties (Choose One Method)

  1. If You Have Radius: Enter radius value in the Radius field. Leave Diameter and Circumference blank.
  2. If You Have Diameter: Enter diameter value in the Diameter field. Leave Radius and Circumference blank.
  3. If You Have Circumference: Enter circumference value in the Circumference field. Leave others blank.
  4. Calculate: Press Calculate button to compute all properties
  5. Review: See radius, diameter, circumference, and area instantly

Area Calculator

  1. Enter Radius OR Diameter: Choose one, leave the other blank
  2. Calculate: Press Calculate to compute area
  3. View Result: See area with step-by-step breakdown

Arc & Sector Calculator

  1. Enter Radius: Provide the circle's radius
  2. Enter Angle: Provide central angle
  3. Select Unit: Choose degrees or radians
  4. Calculate: Find arc length and sector area

Important Tips

  • Only One Value: For basic calculator, enter only ONE known value. Calculator derives all others.
  • Positive Numbers: Radius, diameter, and circumference must be positive
  • Angle Range: Central angle must be between 0° and 360° or 0 to 2π radians
  • Unit Consistency: All measurements use same unit throughout

Complete Formulas Reference

Basic Circle Relationships
D = 2R (Diameter = 2 × Radius)
R = D/2 (Radius = Diameter ÷ 2)
Circumference (Perimeter)
C = 2πR
C = πD
Where π ≈ 3.14159
Area
A = πR²
A = π(D/2)²
A = πD²/4
Arc Length
Arc = (θ/360°) × 2πR (angle in degrees)
Arc = (θ/2π) × 2πR = R × θ (angle in radians)
Sector Area
Sector = (θ/360°) × πR² (angle in degrees)
Sector = (θ/2π) × πR² (angle in radians)

Parts of a Circle

Understanding circle terminology helps identify which parts you need to calculate:

Center (Origin) The point equidistant from all points on the circle. All measurements reference this point.
Radius (R) Distance from center to any point on circle. Foundation for all calculations.
Diameter (D) Longest distance across circle through center. Always equals 2R.
Circumference (C) Distance around circle perimeter. Equals 2πR or πD.
Arc Portion of circle's circumference between two points. Can be major (>half) or minor (
Chord Line segment connecting any two points on circle. Diameter is longest chord.
Sector Area enclosed by two radii and arc. Like pizza slice. "Major" if >180°, "minor" if <180°.
Segment Area between chord and arc. Smaller than sector.
Tangent Line touching circle at exactly one point. Perpendicular to radius at that point.
Secant Line passing through circle at two points. Extension of a chord beyond circle.

The Constant π (Pi)

Pi (π) is the mathematical constant representing the ratio of any circle's circumference to its diameter. Approximately 3.14159, pi is an irrational number—its decimal representation never terminates and never repeats. Ancient mathematicians spent centuries trying to find its exact value.

Pi appears in every circle formula because it fundamentally relates circular measurements. The circumference equals π times the diameter. The area equals π times the radius squared. This constant connects linear measurements (circumference) to area measurements through the circular geometry.

Remarkably, pi is also transcendental—it cannot be expressed as a root of any polynomial with rational coefficients. In 1880, Ferdinand von Lindemann proved this, ending ancient efforts to "square the circle" (construct a square with same area as circle using only compass and straightedge). Despite pi's complexity, its appearance in calculations is simple and elegant.

Pi Facts: π ≈ 3.14159 | π is irrational | π is transcendental | Ratio of circumference to diameter for any circle

Worked Examples

Example 1: Basic Circle from Radius

Problem: Circle with radius 5 cm. Find diameter, circumference, and area.

Solution:
Given: R = 5 cm
D = 2R = 2 × 5 = 10 cm
C = 2πR = 2 × π × 5 ≈ 31.42 cm
A = πR² = π × 5² = π × 25 ≈ 78.54 cm²

Example 2: Circle from Circumference

Problem: Circle with circumference 62.83 meters. Find radius, diameter, and area.

Solution:
Given: C = 62.83 m
C = 2πR, so R = C/(2π) = 62.83/(2π) ≈ 10 m
D = 2R = 20 m
A = πR² = π × 10² ≈ 314.16 m²

Example 3: Circle from Area

Problem: Circle with area 100 square units. Find radius, diameter, and circumference.

Solution:
Given: A = 100
A = πR², so R² = A/π = 100/π ≈ 31.83
R = √31.83 ≈ 5.64 units
D = 2R ≈ 11.28 units
C = 2πR ≈ 35.45 units

Example 4: Arc and Sector (Pizza Slice)

Problem: Pizza (radius 15 cm) cut into 8 equal slices. Find arc length and area of one slice.

Solution:
Central angle = 360°/8 = 45°
Arc length = (45°/360°) × 2πR = (1/8) × 2π × 15 ≈ 11.78 cm
Sector area = (45°/360°) × πR² = (1/8) × π × 15² ≈ 88.36 cm²
Each slice: 11.78 cm arc, 88.36 cm² area

Example 5: Circle from Diameter

Problem: Circular pond with diameter 30 meters. Find radius, circumference, and area.

Solution:
Given: D = 30 m
R = D/2 = 15 m
C = πD = π × 30 ≈ 94.25 m (perimeter)
A = π(D/2)² = π × 15² ≈ 706.86 m² (surface area)

Frequently Asked Questions

❓ What's the difference between circumference and perimeter?
Circumference is the specific term for a circle's perimeter. Perimeter is the general term for distance around any shape. For circles, we use "circumference."
❓ How are radius, diameter, and circumference related?
Diameter = 2 × Radius. Circumference = π × Diameter or 2π × Radius. All three interconnect through these simple relationships.
❓ Why is π approximately 3.14159?
π is the ratio of circumference to diameter. For any circle, circumference ÷ diameter ≈ 3.14159. This is a fundamental mathematical constant, not dependent on size.
❓ What's the difference between arc and sector?
Arc is the curved line (part of circumference). Sector is the area enclosed by two radii and the arc (like pizza slice). Arc measures distance, sector measures area.
❓ How do I find circle area if I only have diameter?
First convert diameter to radius: R = D/2. Then calculate area: A = πR². Or use formula directly: A = π(D/2)² = πD²/4.
❓ What's a chord versus a diameter?
A chord connects any two points on a circle. A diameter is a special chord that passes through the center. Diameter is the longest possible chord.
❓ Can circle calculations use degrees and radians?
Yes. For arc length and sector area, angles can be in degrees or radians. This calculator supports both. 360° = 2π radians.
❓ What applications use circle calculations?
Circle calculations apply to: wheels, pipes, tanks, ponds, pizza, clocks, orbits, architecture, engineering, and countless real-world circular objects and shapes.

Calculate Circle Properties Instantly

Whether you're solving geometry problems, designing circular objects, planning layouts, or analyzing circular patterns, this comprehensive circle calculator handles all circle properties with instant accuracy and step-by-step solutions. Fast, reliable, completely free.

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