Area of a Circle Calculator
Calculate circle area, diameter, circumference, and radius with exact formulas in terms of π, live geometric diagrams, and multi-unit conversions.
Circle Dimensions
cm
cm
cm²
Sector Parameters
cm
deg (°)
Annulus (Ring) Parameters
cm
cm
Circle Area
78.54 cm²
Total Enclosed Surface Area (Exact: 25π cm²)
Radius (r)
5.00 cm
Diameter (d)
10.00 cm
Circumference (C)
31.42 cm
Exact Area (in terms of π)
25π cm²
π × 5²
Circumference (Perimeter)
31.42 cm
10π cm (2 × π × 5)
Bounding Square Area
100.00 cm²
d² = 10 × 10 cm
Circle / Square Coverage
78.54%
π / 4 ≈ 0.785398
Geometric Parameters Summary
Exact Euclidean Values
| Dimension | Decimal Value | Formula / Exact Form |
|---|---|---|
| Radius (r) | 5.0000 cm | r = d / 2 |
| Diameter (d) | 10.0000 cm | 2r |
| Circumference (C) | 31.4159 cm | 2πr = 10π cm |
| Circle Area (A) | 78.5398 cm² | πr² = 25π cm² |
| Bounding Square Area | 100.0000 cm² | d² = 4r² |
2D Geometric Schematic
Enclosed Area (πr²)
Radius Line (r)
Bounding Square (d²)
Area of a Circle Formula & Live Step-by-Step Substitution
A = π × r² = π × (5)² = 25π ≈ 78.5398 cm²
Step 1: Square the radius: 5² = 25.
Step 2: Multiply by Archimedes' constant π (≈ 3.14159265...): 25 × 3.14159265 = 78.5398 cm².
From diameter: A = (π / 4) × d² = (π / 4) × 10² = 0.785398 × 100 = 78.5398 cm².
Step 2: Multiply by Archimedes' constant π (≈ 3.14159265...): 25 × 3.14159265 = 78.5398 cm².
From diameter: A = (π / 4) × d² = (π / 4) × 10² = 0.785398 × 100 = 78.5398 cm².
Equivalent Area in Other Units
Instant Conversion Matrix
| Unit System | Unit | Converted Area |
|---|---|---|
| Metric (SI) | Square Centimeters (cm²) | 78.5398 cm² |
| Metric (SI) | Square Millimeters (mm²) | 7,853.98 mm² |
| Metric (SI) | Square Meters (m²) | 0.007854 m² |
| US Customary | Square Inches (sq in / in²) | 12.1737 in² |
| US Customary | Square Feet (sq ft / ft²) | 0.08454 ft² |
| US Customary | Square Yards (sq yd / yd²) | 0.009393 yd² |
Mathematical Foundations of the Circle
A circle is defined as the set of all coplanar points equidistant from a fixed central point. The ratio of any circle's circumference to its diameter is the universal transcendental mathematical constant π (pi ≈ 3.1415926535...). The area formula was first formally proven rigorously by Archimedes of Syracuse in Measurement of a Circle (c. 260 BC) using the method of exhaustion, proving that the area enclosed by a circle equals that of a right triangle with legs equal to the circle's radius and circumference.
Core Geometric Formulas:
• Area from Radius: A = π × r²
• Area from Diameter: A = (π / 4) × d² ≈ 0.785398 × d²
• Area from Circumference: A = C² / (4π)
• Circumference from Area: C = 2 × √(π × A)
• Radius from Area: r = √(A / π)
• Area from Radius: A = π × r²
• Area from Diameter: A = (π / 4) × d² ≈ 0.785398 × d²
• Area from Circumference: A = C² / (4π)
• Circumference from Area: C = 2 × √(π × A)
• Radius from Area: r = √(A / π)
Sector & Segment Area Calculations
When analyzing portions of a circle, geometry differentiates between sectors and segments:
- Circular Sector (Pie Slice): The region bounded by two radii and the intercepted arc. The area is proportional to the central angle θ:
Asector = (θ / 360°) × πr² = (1/2) × r² × θrad - Arc Length (L): The curved distance along the circumference:
L = (θ / 360°) × 2πr = r × θrad - Circular Segment: The region bounded by a chord and the arc between the chord's endpoints. Its area equals the sector area minus the central triangle area:
Asegment = (1/2) × r² × (θrad − sin(θrad)) - Annulus (Circular Ring): The region between two concentric circles with outer radius R and inner radius r:
Aannulus = π × (R² − r²)
Standard Geometric Benchmark Table
| Radius (r) | Diameter (d) | Circumference (C) | Exact Area (π) | Decimal Area |
|---|---|---|---|---|
| 1 cm | 2 cm | 6.28 cm | 1π cm² | 3.14 cm² |
| 2 cm | 4 cm | 12.57 cm | 4π cm² | 12.57 cm² |
| 5 cm | 10 cm | 31.42 cm | 25π cm² | 78.54 cm² |
| 10 cm | 20 cm | 62.83 cm | 100π cm² | 314.16 cm² |
| 12 in (1 ft) | 24 in (2 ft) | 75.40 in | 144π in² | 452.39 in² (3.14 ft²) |
| 1 m | 2 m | 6.28 m | 1π m² | 3.14 m² |
How to Use This Calculator
- Select Desired Tab: Use "Circle Solver" for standard full circles, "Sector & Segment" for pie slices/chords, or "Annulus (Ring)" for hollow pipes and washers.
- Enter Known Dimension: Type any known value—radius, diameter, circumference, or area—and pick your length unit (cm, m, mm, in, ft, yd).
- Inspect Interactive Visualizer: View the live 2D geometric schematic with labeled radius lines, bounding square outline, and proportional scaling.
- Export Results: Copy the formatted calculation summary to your clipboard or print clean PDF documentation.
Frequently Asked Questions
What is the relationship between the radius and diameter?
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The diameter is exactly twice the length of the radius (d = 2r). Conversely, the radius is half the diameter (r = d / 2). A diameter is the longest chord possible in any circle and passes directly through the center point.
Why does doubling the radius quadruple the circle's area?
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Area scales quadratically with respect to linear dimensions because the formula contains the squared radius term (r²). If you double the radius from r to 2r, the new area becomes π × (2r)² = π × 4r² = 4 × (πr²), which is exactly four times as large.
What is the difference between exact area and decimal area?
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The exact area is expressed in terms of the mathematical constant π (such as 25π cm²), preserving complete infinite precision without rounding. The decimal area replaces π with its numerical approximation (≈ 3.14159), producing a decimal number (78.54 cm²) useful for practical engineering, machining, and construction.
How much of an enclosing bounding square does a circle occupy?
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A circle always covers exactly π / 4 (approximately 78.5398%) of the area of its circumscribed bounding square (where the square side length equals the circle diameter d). The remaining 21.4602% represents the empty corner space outside the circle.
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