SamCalculator

Circle Calculator

Calculate radius, diameter, circumference, area, arc length, sector area, and other circle measurements instantly with step-by-step solutions and interactive visualizations.

Solve every circle property

Enter one value — radius, diameter, circumference, or area — and we'll calculate the other three.

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Circle Formula Library

Every identity used by this calculator, in one place.

Diameter from radius

d = 2r

Diameter is twice the radius.

Radius from diameter

r = d ÷ 2

Radius is half the diameter.

Circumference from radius

C = 2πr

Perimeter of the circle.

Circumference from diameter

C = πd

Same thing — π times the diameter.

Area

A = πr²

Surface enclosed by the circle.

Radius from area

r = √(A ÷ π)

Inverse-square root of area.

Diameter from area

d = √(4A ÷ π)

From A = π(d/2)².

Circumference from area

C = 2√(πA)

Combines C = 2πr and r = √(A/π).

Radius from circumference

r = C ÷ (2π)

Rearrange the circumference formula.

Diameter from circumference

d = C ÷ π

Same identity, expressed via d.

Area from circumference

A = C² ÷ (4π)

Combine the two formulas to skip r.

Arc length

s = rθ (θ in radians)

Or s = (θ/360) × 2πr in degrees.

Sector area

A_sec = (θ/360) × πr²

Slice of the pie at central angle θ.

Chord length

c = 2r·sin(θ/2)

Straight line across the arc.

Sagitta

h = r − r·cos(θ/2)

Arc height — from chord to arc.

Segment area

A_seg = ½r²(θ − sinθ)

Sector minus the triangle.

Annulus area

A_ring = π(R² − r²)

Doughnut — outer minus inner.

Area Comparison Across Sample Radii

Area = πr² rises sharply with the radius.

What Is a Circle?

A circle is the set of all points in a plane that lie the same distance — the radius — from a fixed centre. Every circle is described completely by a single number: the radius. From that one value, every other property is fixed. The diameter is 2r; the circumference is 2πr; the area is πr². Knowing any one of those four values is enough to recover the other three.

This calculator bundles five tools on a single page: a Circle Calculator that fills in every property from any one input, an Arc Length Calculator for partial perimeters, a Sector Area Calculator for pie-slice areas, a Chord Calculator for straight-line cuts, and a Circle Geometry Toolkit with converters, ring areas, scaling, and circle packing. Pair it with the scientific calculator, the slope calculator, and the unit converter for full coordinate-geometry workflows.

How Circle Calculations Work

Everything reduces to the radius

Each of the four primary measurements — radius, diameter, circumference, area — can be converted into the radius first, then the other three are derived. Diameter halves; circumference divides by 2π; area takes the square root of A/π. After that, d = 2r, C = 2πr, and A = πr² do the rest.

Arcs and sectors scale with the angle

Both the arc length and the sector area are proportional to the central angle θ. An arc covering θ degrees is (θ/360) of the circumference; a sector with central angle θ is (θ/360) of the disk area. Switching to radians is even cleaner: s = rθ and A_sec = ½r²θ.

Chords use trigonometry

A chord is the straight line connecting two points on a circle. Its length is c = 2r·sin(θ/2) — derived by dropping a perpendicular from the centre to the chord. The chord, the two radii, and the apothem r·cos(θ/2) form an isosceles triangle, with the sagitta r − r·cos(θ/2) being the bulge of the arc above the chord.

Area scales with the square of the radius

Every linear dimension of a circle (radius, diameter, chord, arc) scales linearly with a chosen factor — but area scales with the square. Doubling the radius quadruples the area; tripling it nines the area. This is the same square law that governs pizza prices, satellite-dish gain, and inverse-square physics.

6 Ways to Use This Circle Calculator

1

Find every property from any one input

Enter the radius, diameter, circumference, or area — the calculator returns the other three, plus radius², diameter², and π in one pass.

2

Compute arc length for a given angle

Use the Arc tab to find the length of a curved boundary — useful for cable bends, road curves, and clock hand sweeps.

3

Calculate pie-slice sector areas

Enter the radius and central angle to find a sector — the math behind pie charts, pizza slices, and theatre seating wedges.

4

Measure chords, sagittae and apothems

Use the Chord tab to find straight-line distances across a circle — essential for arch construction, bridge design, and gear teeth.

5

Estimate ring or doughnut areas

The Toolkit annulus mode subtracts an inner disk from an outer disk — used for washers, gaskets, planetary orbital bands, and CD/DVD geometry.

6

Scale a circle up or down

The Toolkit scale mode multiplies the radius by a factor and shows linear, circumferential, and quadratic area changes side by side.

Best Practices for Working With Circles

Mind the units. Linear measurements (radius, diameter, circumference, arc length, chord) share the same unit — metres, inches, kilometres, light-years, whatever you picked. Area is always in the square of that unit (m², in², ft²). Mixing 1 m with 50 cm will give you nonsense; convert first using the unit converter.

Pick the right angle unit. Pure mathematics prefers radians — the formulas s = rθ and A_sec = ½r²θ work only in radians. Engineering, navigation, and everyday speech usually use degrees. This calculator accepts both; just keep your inputs consistent with the formulas you're using.

Don't round π too early. π is irrational — using 3.14 instead of π introduces a 0.05% error, which adds up over many steps. The calculator uses the JavaScript Math.PI constant (≈15 digits), then formats the output to a friendly number of decimals.

Why Circle Math Matters

Engineering and architecture

Cylindrical pipes, drum brakes, gear teeth, structural columns, domes, arches, and roundabouts all depend on circle math. Hoop stress, hydraulic flow, and material take-off lists all start with πr² and 2πr.

Physics and astronomy

Orbits, electron shells, lens optics, and most rotational mechanics are first analysed assuming circles. Kepler's first law upgraded planetary paths from circles to ellipses, but circular approximations remain a useful starting point in nearly every physical system.

Manufacturing and machining

CNC cutters, lathe tooling, drill bits, and turn-mill operations all operate on circular paths. Tolerance stack-up, surface speed (Vc = πd·n), and arc tool-paths are calculated from these formulas thousands of times a day in modern shops.

Geography and navigation

Great-circle distance, longitude lines, GPS satellite orbits, and time-zone widths are circular by construction. Earth's circumference (≈ 40 075 km at the equator) is the foundation of every map projection.

Where Circle Math Gets Tricky

Reflex angles

A central angle larger than 180° is called reflex. Arc-length and sector-area formulas still hold up to 360°, but visualisations differ — the 'minor' and 'major' arcs swap roles.

Degrees vs radians

π/3 radians is 60°, not 60. The formula s = rθ assumes θ is in radians; using degrees overestimates by a factor of 180/π ≈ 57.3. Always convert before substituting.

Diameter or radius?

The single most common error in circle problems is confusing the two. The diameter is the chord through the centre; the radius is half of that. C = πd is correct, but C = πr is wrong by a factor of two.

Numerical precision

π is irrational. For most physical problems, double-precision (≈15 significant digits) is more than enough. But chained calculations — turning C into r into A — can accumulate rounding error; keep the radius as your single source of truth.

Core Circle Formulas

Diameter

d = 2r

Twice the radius.

Radius from diameter

r = d ÷ 2

Half the diameter.

Circumference

C = 2πr = πd

Two equivalent forms.

Area

A = πr²

Defining area identity.

Radius from area

r = √(A ÷ π)

Inverse of A = πr².

Circumference → area

A = C² ÷ (4π)

Skip the radius step.

Arc length

s = rθ (rad) = (θ/360)·2πr

Two angle units, same answer.

Sector area

A_sec = ½r²θ = (θ/360)·πr²

Slice of the disk.

Chord length

c = 2r·sin(θ/2)

Straight line across the arc.

Apothem

m = r·cos(θ/2)

Centre to chord, perpendicular.

Sagitta

h = r − r·cos(θ/2)

Bulge of arc above chord.

Segment area

A_seg = ½r²(θ − sin θ)

Sector minus triangle.

Annulus area

A_ring = π(R² − r²)

Outer disk minus inner.

Area scaling

A' ÷ A = k²

Square law — area scales with k².

Circle constant

π = C ÷ d ≈ 3.14159…

Irrational, transcendental.

Common Circle Mistakes

  1. 1

    Using diameter where the formula expects radius

    C = 2πr but C = πd; A = πr² but A = π(d/2)² = πd²/4. Substituting the diameter into a radius formula doubles the linear answer and quadruples the area answer.

  2. 2

    Forgetting to switch units when computing area

    A radius in centimetres gives an area in cm², not in cm. Always note the squared unit on every area result; mixing m² with cm² is the most common engineering arithmetic mistake.

  3. 3

    Using degrees in s = rθ

    The formula s = rθ assumes θ is in radians. Convert first: θ_rad = θ_deg × π/180. Or use s = (θ/360) × 2πr if you prefer to stay in degrees.

  4. 4

    Confusing arc length with chord length

    The arc is the curved path along the circle; the chord is the straight line through the disk. They're only equal in the limit of θ → 0; for any finite angle the arc is longer.

  5. 5

    Treating the circle constant as exactly 3.14

    Using 3.14 introduces a ≈ 0.05% error. For school answers that's fine; for engineering tolerances it can be significant. Use π directly whenever possible.

  6. 6

    Forgetting the square law on area

    Doubling the radius does not double the area — it quadruples it. A 12-inch pizza has four times the topping of a 6-inch pizza, despite costing only twice as much: a real value comparison.

Parts of a Circle

Centre and radius

Every circle has a centre point and a radius — the constant distance from centre to perimeter. Some textbooks call the centre the 'origin' of the circle.

Diameter

A chord that passes through the centre. The longest possible chord, equal to 2r. Splitting any diameter in half recovers the radius.

Arc

Any continuous portion of the circumference. A minor arc is less than half the circle; a major arc is more than half.

Chord

A straight line whose endpoints lie on the circle. The longest chord is the diameter; the shortest is the perpendicular from the centre to the chord (the apothem).

Sector

A pie-slice region bounded by two radii and an arc. Sector area equals ½r²θ when θ is in radians.

Segment, sagitta, apothem

A segment is the region between a chord and its arc. The sagitta is the perpendicular distance from chord to arc; the apothem is the perpendicular distance from centre to chord.

Real-Life Uses of Circle Math

Cooking and baking

Comparing pizza sizes, scaling cake-tin areas, working out the volume of a cylindrical pot — all start with πr². A 9-inch cake tin holds 1.27× the batter of an 8-inch tin (square law).

Construction

Quantity surveying for round columns, domes, manhole covers, drainage pipes, and circular foundations relies on circumference and area formulas. Hoop reinforcement is laid out per metre of arc.

Space and orbits

Low Earth orbit at ~7,000 km radius gives a circumference of ~44,000 km — that's how far a satellite travels per revolution, and it's the same calculation as the circumference of the Earth, just bigger.

Optics and antennas

Lens aperture area, satellite-dish gain, and radio-telescope effective area all scale with πr². Doubling a dish's diameter quadruples its collecting area — and roughly quadruples its sensitivity.

Circle Formula Cheat Sheet

QuantityFormula
Radiusr = d ÷ 2 = C ÷ (2π) = √(A ÷ π)
Diameterd = 2r = C ÷ π
CircumferenceC = 2πr = πd
AreaA = πr² = πd² ÷ 4 = C² ÷ (4π)
Arc lengths = rθ (θ in radians)
Sector areaA_sec = (θ ÷ 360) × πr² = ½r²θ
Chord lengthc = 2r·sin(θ/2)
Sagittah = r − r·cos(θ/2)
Segment areaA_seg = ½r²(θ − sin θ)
Annulus areaA_ring = π(R² − r²)

Methodology you can verify

Every value is computed live in your browser with the JavaScript Math.PI constant (≈15 significant digits), classical algebraic identities, and trigonometric identities for chords and sagittae. No numbers are hard-coded. Read more on the methodology and editorial policy pages.

Frequently Asked Questions

Start with the area formula A = πr² and solve for the radius: r = √(A ÷ π). Once you have the radius, every other property follows — diameter d = 2r, circumference C = 2πr, and you already have A. For example, with A = 100 the radius is √(100 ÷ π) ≈ 5.6419, the diameter is ≈ 11.2838, and the circumference is ≈ 35.4491. The Circle tab on this page does exactly this calculation and shows the full step-by-step working below the result.

Multiply the radius by 2π — the circumference is C = 2πr. With r = 5 the circumference is 2 × π × 5 ≈ 31.4159. Equivalently, you can multiply the diameter by π: C = πd. Both formulas give the same answer because d = 2r. The Circle tab fills both forms in automatically once you enter any one measurement; the Step-by-Step Solution shows each substitution.

Double the radius — d = 2r — or, if you only know the circumference, use d = C ÷ π. If you have the area, the diameter is d = √(4A ÷ π) = 2√(A ÷ π). Every one of these formulas yields the same diameter because each property is just a different lens onto the same underlying radius. The Circle tab handles all four input modes interchangeably.

The area of a circle is A = πr², where r is the radius and π ≈ 3.14159. If you only have the diameter, use A = πd² ÷ 4. If you only have the circumference, use A = C² ÷ (4π). All three are algebraically equivalent. Area scales with the square of the radius — doubling r quadruples A — which is the classic 'square law' behind everything from pizza pricing to satellite-dish gain.

Arc length is s = rθ when θ is in radians, or s = (θ ÷ 360) × 2πr when θ is in degrees. Both forms give the same answer. For r = 10 and θ = 90°, the arc length is (90 ÷ 360) × 2π × 10 = (1 ÷ 4) × 20π ≈ 15.708. The Arc tab on this page accepts both degree and radian inputs and converts between them automatically. Arc length is what you would measure if you wrapped a flexible tape around the curved boundary.

A sector is a pie-slice region of a disk bounded by two radii and the arc between their endpoints. The sector area is (θ ÷ 360) × πr² when θ is in degrees, or A_sec = ½r²θ when θ is in radians. For r = 10 and θ = 120°, the sector area is (120 ÷ 360) × π × 10² ≈ 104.72. The Sector tab also returns the remaining area (the rest of the disk), the sector's arc length, perimeter (arc + 2r), and the area of the underlying triangular wedge as a 'segment area'.

Chord length is c = 2r·sin(θ/2), where θ is the central angle subtended by the chord. For r = 10 and θ = 60°, the chord is 2 × 10 × sin(30°) = 20 × 0.5 = 10. Interestingly, the chord equals the radius at exactly 60° — a fact that makes regular hexagon construction trivial with only a compass and straightedge. The Chord tab also returns the sagitta (chord-to-arc bulge), the apothem (centre-to-chord distance), and the segment area.

Circumference is the perimeter of a circle — the distance you would travel going once around its boundary. For a circle of radius r, circumference C = 2πr ≈ 6.2832 r. For a unit circle (r = 1) the circumference is exactly 2π. Earth's equatorial circumference is roughly 40,075 km. Circumference scales linearly with the radius, unlike area which scales with the square of the radius.

π is the universal constant that converts a circle's diameter into its circumference: C = πd. It appears in every circle and sphere formula — circumference (C = 2πr), area (A = πr²), surface area of a sphere (4πr²), and volume of a sphere (⁴⁄₃ πr³). π is irrational (no exact decimal or fraction) and transcendental (not the root of any algebraic equation). This calculator uses the JavaScript Math.PI constant (≈15 significant digits) so you never have to remember more than a few decimals yourself.

Circle math is used in engineering (pipes, gears, drums, columns), architecture (domes, arches, roundabouts, columns), manufacturing (drill bits, lathes, CNC tooling), navigation (great-circle distances, GPS orbits), astronomy (planetary orbits, satellite orbits, telescope apertures), physics (rotational mechanics, optics), construction (round foundations, drainage pipes), and everyday tasks like comparing pizza sizes or cake-tin volumes. Anywhere a curve, hole, wheel, or revolution appears, πr², 2πr, and arc length s = rθ are the working tools.

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Pair the circle calculator with other math tools for trigonometry, slopes, ratios, and unit conversions.