Circle Calculator
Calculate circle area, circumference, diameter, and radius instantly. Includes arc length and sector area calculators with visual diagram.
Enter Any Known Value
Enter any circle measurement above to calculate all properties.
Quick Reference:
Radius (r): Center to edge | Diameter (d): Across through center
Circumference (C): Distance around | Area (A): Space inside
Related Calculators
About This Calculator
The Circle Calculator instantly computes all circle measurements from any single known value. Enter the radius, diameter, circumference, or area, and the calculator automatically finds all other values using precise mathematical formulas based on pi (π). Circles are fundamental geometric shapes found everywhere in nature and human design—from wheels and coins to planetary orbits and pizza slices. Understanding circle mathematics is essential for countless real-world applications including construction (circular patios and pools), engineering (pipes and gears), science (atomic structure and wave mechanics), and everyday tasks like determining how much fence you need around a circular garden. Our calculator goes beyond basic conversions with advanced features for arc length and sector area calculations, perfect for tasks like measuring curved walkways or calculating the area of a pie slice. Whether you're a student learning geometry, an engineer designing circular components, a contractor estimating materials, or simply curious about circle math, this comprehensive tool provides instant, accurate results with step-by-step formulas shown. Enter any measurement to discover the elegant mathematical relationships that govern all circles—where the ratio of circumference to diameter always equals pi, one of mathematics' most famous constants.
How to Use the Circle Calculator
- 1Select which measurement you know: Radius, Diameter, Circumference, or Area.
- 2Enter your known value in the input field with the appropriate unit.
- 3All other circle measurements are calculated automatically in real-time.
- 4View the visual diagram showing your circle with labeled dimensions.
- 5Toggle Advanced Mode to access arc length and sector area calculators.
- 6For arc length, enter the central angle in degrees to find the curved distance.
- 7For sector area, the angle determines what fraction of the total area is calculated.
- 8Use the precision toggle to display more or fewer decimal places as needed.
Formula
A = πr², C = 2πr, d = 2rWhere r is the radius, d is the diameter, C is the circumference, A is the area, and π (pi) is approximately 3.14159. The radius is the fundamental measurement from which all others derive.
Essential Circle Formulas
The Four Core Circle Measurements:
Every circle can be completely described by four related measurements. Knowing any one allows you to calculate all others:
1. Radius (r) The distance from the center to any point on the circle's edge. This is the fundamental measurement from which all others derive.
2. Diameter (d) The distance across the circle through its center. Always exactly twice the radius.
- Formula: d = 2r
- Inverse: r = d / 2
3. Circumference (C) The distance around the circle (its perimeter). This is where pi enters the equations.
- Formula: C = 2πr = πd
- Inverse: r = C / (2π)
4. Area (A) The space enclosed within the circle, measured in square units.
- Formula: A = πr²
- Inverse: r = √(A / π)
Quick Reference Table:
| Known Value | Find Radius | Find Diameter | Find Circumference | Find Area |
|---|---|---|---|---|
| Radius (r) | r | 2r | 2πr | πr² |
| Diameter (d) | d/2 | d | πd | π(d/2)² |
| Circumference (C) | C/(2π) | C/π | C | π(C/(2π))² |
| Area (A) | √(A/π) | 2√(A/π) | 2π√(A/π) | A |
Understanding Pi (π)
What is Pi?
Pi (π) is the ratio of any circle's circumference to its diameter. This ratio is the same for ALL circles, regardless of size—from the smallest atom to the largest galaxy.
Pi's Value: π = 3.14159265358979323846...
Pi is an irrational number, meaning its decimal expansion never ends and never repeats. For most calculations:
- Quick estimate: 3.14
- Better precision: 3.14159
- Calculator precision: 3.141592653589793
Pi Approximations Through History:
| Civilization | Approximation | Accuracy |
|---|---|---|
| Ancient Babylon | 3.125 | 0.5% error |
| Ancient Egypt | 3.1605 | 0.6% error |
| Archimedes | 3.1418 | 0.002% error |
| Chinese (Zu Chongzhi) | 355/113 = 3.1415929 | 0.000008% error |
| Modern computers | Trillions of digits | Exact (computational limit) |
Common Pi Fractions:
- 22/7 = 3.142857... (0.04% error)
- 355/113 = 3.1415929... (extremely accurate!)
- 333/106 = 3.1415094...
Why Pi Matters: Pi appears throughout mathematics and physics, not just in circles:
- Wave equations and oscillations
- Probability and statistics (normal distribution)
- Complex numbers (Euler's identity: e^(iπ) + 1 = 0)
- Quantum mechanics and relativity
Arc Length and Sector Area
Arc Length Formula:
An arc is a portion of the circumference. Its length depends on the central angle.
Formula: Arc Length = (θ / 360°) × 2πr
Or in radians: Arc Length = θ × r
Examples:
- Quarter circle (90°): Arc = (90/360) × 2πr = πr/2
- Half circle (180°): Arc = (180/360) × 2πr = πr
- Full circle (360°): Arc = 2πr (the circumference)
Sector Area Formula:
A sector is a "pie slice" of the circle. Its area depends on the central angle.
Formula: Sector Area = (θ / 360°) × πr²
Or in radians: Sector Area = (θ / 2) × r²
Examples:
- Quarter circle (90°): Sector = (90/360) × πr² = πr²/4
- Half circle (180°): Sector = (180/360) × πr² = πr²/2
- Pizza slice (45°): Sector = (45/360) × πr² = πr²/8
Practical Applications:
- Curved walkways and driveways
- Pizza/pie portion sizes
- Fan blade areas
- Windshield wiper coverage
- Protractor markings
Solving for Any Circle Value
From Radius: This is the easiest starting point—all formulas use radius directly.
- Diameter: d = 2r
- Circumference: C = 2πr
- Area: A = πr²
From Diameter: Divide by 2 to get radius, then calculate others.
- Radius: r = d / 2
- Circumference: C = πd
- Area: A = π(d/2)² = πd²/4
From Circumference: Involves dividing by pi to "undo" the circumference formula.
- Radius: r = C / (2π)
- Diameter: d = C / π
- Area: A = C² / (4π)
From Area: Requires taking a square root to "undo" the squared radius.
- Radius: r = √(A / π)
- Diameter: d = 2√(A / π)
- Circumference: C = 2π√(A / π) = 2√(πA)
Step-by-Step Example: Given: Circumference = 31.4 inches Find: All other measurements
- Radius = 31.4 / (2π) = 31.4 / 6.283 = 5 inches
- Diameter = 2 × 5 = 10 inches
- Area = π × 5² = π × 25 = 78.54 square inches
Real-World Circle Applications
Construction and Home Improvement:
Calculating materials for circular projects is essential for accurate estimates.
- Circular Patios: Area determines concrete/paver quantity
- Round Pools: Circumference for edging, area for covers
- Circular Gardens: Perimeter for fencing, area for mulch/soil
- Circular Windows: Diameter for ordering, area for glass quantity
Related: Concrete Calculator - Calculate cubic yards for circular pads Related: Paint Calculator - Estimate paint for circular surfaces
Engineering and Manufacturing:
- Pipes and Tubes: Cross-sectional area determines flow rate
- Wheels and Gears: Circumference determines travel distance per rotation
- Circular Saw Blades: Diameter determines cutting depth
- Bearings: Precise radius for proper fit
Science and Nature:
- Planetary Orbits: Approximately circular paths
- Tree Rings: Cross-sectional area indicates growth
- Ripples in Water: Expanding circles demonstrate wave behavior
- Atomic Structure: Electron orbital diagrams
Everyday Applications:
- Pizza Sizing: Diameter determines area (and value!)
- Clock Faces: Circumference for spacing hour markers
- Coins: Diameter for identification, area for metal content
- Hula Hoops: Circumference for material length
Related: Area Calculator - General area calculations for all shapes Related: Volume Calculator - Calculate sphere and cylinder volumes
Circle Properties and Theorems
Fundamental Properties:
-
All points equidistant: Every point on a circle is exactly the same distance (radius) from the center.
-
Constant ratio: Circumference / Diameter = π for ALL circles.
-
Symmetry: A circle has infinite lines of symmetry (any diameter).
-
Maximum area: Among all shapes with the same perimeter, a circle encloses the maximum area.
Important Circle Theorems:
Inscribed Angle Theorem: An inscribed angle is half the central angle subtending the same arc.
Thales' Theorem: Any angle inscribed in a semicircle is a right angle (90°).
Chord-Chord Theorem: When two chords intersect inside a circle, the products of their segments are equal.
Tangent Properties:
- A tangent line touches the circle at exactly one point
- The tangent is perpendicular to the radius at the point of tangency
- Two tangent lines from an external point are equal in length
Circle Constant Relationships:
- C = πd (circumference to diameter)
- A = (C²) / (4π) (area from circumference)
- C = 2√(πA) (circumference from area)
Circles in Coordinate Geometry
Standard Form Equation:
A circle with center (h, k) and radius r: (x - h)² + (y - k)² = r²
Examples:
- Circle at origin with radius 5: x² + y² = 25
- Circle at (3, 4) with radius 2: (x - 3)² + (y - 4)² = 4
General Form: x² + y² + Dx + Ey + F = 0
Where:
- Center: (-D/2, -E/2)
- Radius: √((D² + E²)/4 - F)
Parametric Equations: For a circle of radius r centered at origin:
- x = r cos(θ)
- y = r sin(θ)
Where θ ranges from 0 to 2π.
Converting Forms: From general to standard: Complete the square for x and y terms.
Example: x² + y² - 6x + 4y - 12 = 0
- Group: (x² - 6x) + (y² + 4y) = 12
- Complete squares: (x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
- Standard form: (x - 3)² + (y + 2)² = 25
- Center: (3, -2), Radius: 5
Unit Conversions for Circles
Linear Measurements (Radius, Diameter, Circumference):
| From | To Inches | To Feet | To Meters | To CM |
|---|---|---|---|---|
| 1 inch | 1 | 0.0833 | 0.0254 | 2.54 |
| 1 foot | 12 | 1 | 0.3048 | 30.48 |
| 1 meter | 39.37 | 3.281 | 1 | 100 |
| 1 cm | 0.3937 | 0.0328 | 0.01 | 1 |
Area Measurements:
| From | To sq in | To sq ft | To sq m | To sq cm |
|---|---|---|---|---|
| 1 sq inch | 1 | 0.00694 | 0.000645 | 6.452 |
| 1 sq foot | 144 | 1 | 0.0929 | 929 |
| 1 sq meter | 1550 | 10.764 | 1 | 10,000 |
| 1 sq cm | 0.155 | 0.00108 | 0.0001 | 1 |
Important Conversion Note: When converting area, square the linear conversion factor!
Example: Circle with radius 2 feet
- Area = π(2)² = 12.57 sq ft
- In sq inches: 12.57 × 144 = 1,810 sq in
- In sq meters: 12.57 × 0.0929 = 1.17 sq m
Common Circle Sizes:
| Object | Typical Diameter | Circumference | Area |
|---|---|---|---|
| Quarter | 0.955 in | 3.0 in | 0.72 sq in |
| CD/DVD | 120 mm | 377 mm | 11,310 sq mm |
| Basketball | 9.43 in | 29.6 in | 69.9 sq in |
| Bicycle wheel (700c) | 27.5 in | 86.4 in | 594 sq in |
| Pizza (Large) | 14 in | 44 in | 154 sq in |
Circle vs Other Shapes Comparison
Why Circles are Special:
Maximum Area for Given Perimeter: A circle encloses more area than any other shape with the same perimeter.
Comparison for perimeter = 12 units:
- Circle: Area = 11.46 sq units
- Square: Area = 9 sq units
- Triangle (equilateral): Area = 6.93 sq units
Minimum Perimeter for Given Area: A circle requires the shortest perimeter to enclose a given area.
Comparison for area = 100 sq units:
- Circle: Perimeter = 35.4 units
- Square: Perimeter = 40 units
- Rectangle (2:1): Perimeter = 42.4 units
This is why:
- Bubbles are spherical (minimum surface area)
- Manhole covers are circular (can't fall through the hole)
- Pipes use circular cross-sections (maximum flow for material used)
- Cell membranes tend toward circular shapes
Circle-Square Relationships: For a circle inscribed in a square (touching all sides):
- Circle diameter = Square side
- Circle area = π/4 × Square area ≈ 78.5% of square
For a circle circumscribing a square (passing through corners):
- Circle diameter = Square diagonal = side × √2
- Square area = 2/π × Circle area ≈ 63.7% of circle
Common Circle Calculations with Examples
Example 1: Circular Patio Planning a circular patio with diameter 12 feet.
- Radius = 6 feet
- Area = π × 6² = 113.1 sq ft (for pavers/concrete)
- Circumference = π × 12 = 37.7 feet (for edging)
Example 2: Pizza Value Comparison Which is a better deal: 12" pizza for $12 or 16" pizza for $18?
- 12" pizza: Area = π × 6² = 113.1 sq in, Price = $0.106/sq in
- 16" pizza: Area = π × 8² = 201.1 sq in, Price = $0.090/sq in
- 16" pizza is 15% better value!
Example 3: Round Pool Cover Pool diameter: 18 feet, need 1 foot overlap.
- Cover diameter needed: 20 feet
- Cover radius: 10 feet
- Cover area: π × 10² = 314.2 sq ft
Example 4: Wheel Rotation Distance Bicycle wheel diameter: 26 inches. How far does it travel in one rotation?
- Circumference = π × 26 = 81.7 inches = 6.8 feet
- 100 rotations = 680 feet
Example 5: Circular Garden Fence Garden radius: 8 feet. How much fence is needed?
- Circumference = 2π × 8 = 50.3 feet
- Add 10% for gates/overlap: 55 feet of fencing
Example 6: Pipe Flow Area Inner diameter: 4 inches. What's the cross-sectional area?
- Radius = 2 inches
- Area = π × 2² = 12.57 sq inches
Pro Tips
- 💡Always identify which measurement you know (radius, diameter, circumference, or area) before calculating.
- 💡Remember: diameter = 2 × radius. Many real-world measurements give diameter, but formulas often need radius.
- 💡For quick mental math, circumference is roughly 3 times the diameter (exact: 3.14159...).
- 💡Area increases with the SQUARE of the radius—doubling the radius quadruples the area.
- 💡When comparing pizza sizes, a 16" pizza has 78% more area than a 12" pizza, not 33% more!
- 💡Arc length and sector area use the same angle fraction: angle/360° of the total.
- 💡For circular patios or pools, order 5-10% extra material to account for cutting waste.
- 💡Use 22/7 for pi when doing mental math with numbers divisible by 7.
- 💡The largest circle that fits in a square has diameter equal to the square's side.
- 💡A circle has the maximum area for any given perimeter—nature uses circles for efficiency.
- 💡When measuring round objects, wrap a string around for circumference, then calculate radius.
- 💡For construction projects, always measure diameter in multiple directions to verify the circle is truly round.
Frequently Asked Questions
To calculate circle area, use the formula A = πr² (pi times radius squared). If you know the diameter, first divide by 2 to get the radius, then square it and multiply by pi (3.14159...). For example, a circle with radius 5 has area = π × 25 = 78.54 square units. Our calculator handles this instantly—just enter any known value.
