Circle Calculator
Four modes: full circle (area, circumference, radius, diameter from any one input), arc length, sector area, and annulus area. Enter any known value — get all others instantly.
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How It Works
Enter any ONE of: radius, diameter, circumference, or area. The calculator derives the other three instantly. Formulas: Area=πr², C=2πr, d=2r. Enter diameter 20: r=10, C=62.832, Area=314.159. Enter area 100: r=√(100/π)=5.642, d=11.284, C=35.449.
A=πr² | C=2πr | d=2r | r=√(A/π)Enter r=7 → A=153.938, C=43.982, d=14Enter radius and central angle (degrees). Arc length = (θ/360) × 2πr. The chord length (straight line between endpoints) and angular measure in radians also display. Example: r=10, θ=45°: arc=7.854, chord=7.654, radians=0.785.
L = (θ/360°) × 2πr | chord = 2r×sin(θ/2)r=10, θ=45° → arc=7.854, chord=7.654Enter radius and central angle. Sector area = (θ/360) × πr². Arc length of the sector boundary also displays. A full circle (θ=360°) gives the complete circle area. Quarter circle (θ=90°) gives πr²/4. For a 60° sector with r=12: area=75.398, arc=12.566.
Sector area = (θ/360°) × πr² | arc = (θ/360°) × 2πrr=12, θ=60° → area=75.398, arc=12.566Enter outer radius R and inner radius r. Annulus area = π(R²−r²). Outer circumference = 2πR. Inner circumference = 2πr. The area formula factors as π(R+r)(R−r). For a pipe with outer radius 8 and inner radius 5: area = π(64−25) = 39π ≈ 122.52 cm².
A = π(R²−r²) = π(R+r)(R−r) | R > r > 0R=8, r=5 → area=122.52, outer C=50.265π ≈ 3.14159265... is irrational and transcendental — it cannot be expressed as a fraction or as the root of any polynomial. Every circle formula derives from π: area=πr², circumference=2πr, sphere volume=(4/3)πr³. The calculator uses full floating-point π (15+ digits) for maximum precision.
π = C/d for ANY circle | π ≈ 3.14159265358979C=31.4159..., d=10 → C/d=π exactlyQuick Reference
Verify these in the calculator above.
Circle
Area, r=5
78.540
Circle
Circumference, r=5
31.416
Circle
Radius from area=100
5.642
Arc
Arc: r=10, θ=90°
15.708
Sector
Sector: r=6, θ=90°
area=28.274
Sector
Quarter circle area, r=4
12.566
Annulus
Annulus: R=10, r=6
area=201.06
Sector
Full circle as sector
θ=360°
Tips & Shortcuts
In Circle mode, you can enter any one of the four values — radius, diameter, circumference, or area. The calculator always solves for the other three.
For pie chart sections: use Sector mode with the percentage converted to degrees (multiply by 3.6). A 25% slice = 90°.
Annulus area equals the area of the outer circle minus the area of the inner circle: π×R² − π×r² = π(R²−r²).
Arc length in radians: L = r × θ (where θ is in radians). The Arc mode shows the radian measure alongside degrees automatically.
For a semicircle: use Sector mode with angle 180°, or divide Circle mode area and circumference by 2 (then add the diameter for the full perimeter).
Common Mistakes
Confusing radius and diameter
Radius = half the diameter. Area = πr² NOT πd². The most common error is using diameter directly in the area formula: π×10² = 314 but that is for r=10, not d=10. For d=10: area = π×5² = 78.54.
Using degrees when radians are needed (or vice versa)
The Arc and Sector modes accept degrees. For radians: the calculator shows the radian equivalent. To convert: radians = degrees × π/180. A 90° sector = π/2 radians ≈ 1.5708.
Calculating annulus with inner radius larger than outer
R must be greater than r. If R=5 and r=8, the annulus has no area — the inner circle would extend beyond the outer. Always enter the larger radius as R.
Expecting circumference in square units
Circumference is a length — it has the same units as the radius (e.g. cm, m, inches). Area is in square units (cm², m²). They are fundamentally different quantities.
Using π as 3.14 for precise calculations
π ≈ 3.14 is only accurate to 0.05%. For construction or engineering, use at least 3.14159 (0.0001% error) or use this calculator which applies full precision automatically.
Frequently Asked Questions
Area = π × r² = π × 25 ≈ 78.5398 square units. Circumference = 2 × π × 5 ≈ 31.4159 units. Diameter = 10. Enter 5 in the radius field in Circle mode to get all four values instantly.
Radius = √(Area / π). If area = 50: radius = √(50/π) = √(15.915) ≈ 3.989. In Circle mode, enter the area value — the calculator solves for radius, diameter, and circumference automatically. You can enter any one of the four values and get the other three.
Arc length is the curved distance along a portion of the circumference. Formula: L = (θ/360°) × 2πr where θ is the central angle in degrees. For r=10 and θ=60°: L = (60/360) × 2π × 10 = π × 10/3 ≈ 10.472. Switch to Arc mode and enter radius and angle.
A sector is a pie-slice region bounded by two radii and an arc. Area = (θ/360°) × πr². For a quarter circle (θ=90°) with r=6: area = (90/360) × π × 36 = 9π ≈ 28.274. Switch to Sector mode, enter radius and angle. The arc length of the sector also displays automatically.
An annulus is the region between two concentric circles — a ring or washer shape. Area = π(R² − r²) = π(R+r)(R−r) where R is the outer radius and r the inner radius. Example: R=10, r=6: area = π(100−36) = 64π ≈ 201.06. The outer and inner circumferences also display in Annulus mode.
The calculator uses JavaScript's Math.PI = 3.141592653589793... (full floating-point precision, 15-16 significant digits). Results display to 6 significant figures, which is accurate to within 0.0001% — more than enough for any engineering or geometry calculation.
Circumference = π × diameter, always. This ratio C/d = π is literally the definition of π — it holds for every circle regardless of size. Given circumference C: diameter = C/π, radius = C/(2π). Enter circumference in Circle mode to find radius and area from it.
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