Circle Calculator | Area & Circumference

Calculate radius, diameter, circumference, and area of a circle from any known value. Supports sector angle input for arc length, sector area, chord length, and segment area. Includes 7 unit options, adjustable precision, SVG diagram, and 5 real-world presets.

Quick Presets

Known Measurement

Radius

Unit

Sector Angle(optional, for arc, sector & chord properties)

degrees (°)

Decimal Places

Enter calculate · Esc reset

What Is the Circle Calculator | Area & Circumference?

A circle is defined as the set of all points in a plane that are equidistant from a fixed centre point. That distance is the radius (r). Every other measurement of a circle, diameter, circumference, and area, follows directly from the radius through simple formulas involving π.

What makes this calculator particularly useful is that you don't need to know the radius first. Engineers often measure a pipe's circumferencewith a tape measure; architects work from a circle's area; machinists specify diameters. Enter whichever measurement you have, and every other property is computed instantly.

The optional sector angle field unlocks arc length, sector area, chord length, and chord segment area, covering pie-chart design, gear tooth geometry, stadium seating layouts, and any other problem involving a slice of a circle.

Formula

Core Formulas, Given Radius r

Diameter: d = 2r

Circumference: C = 2πr ≈ 6.28318 × r

Area: A = πr² ≈ 3.14159 × r²

Solving for Radius from Any Known Property

From diameter: r = d / 2

From circumference: r = C / (2π)

From area: r = √(A / π)

Sector and Arc (angle θ in degrees)

Arc length = (θ / 360) × 2πr = θ × πr / 180

Sector area = (θ / 360) × πr²

Chord length = 2r × sin(θ / 2) (where θ is the central angle)

Segment area = (r² / 2)(θ_rad − sin θ_rad) (θ in radians)

All Formulas at a Glance

Property	Primary formula	Alternative
Radius r	d / 2	√(A / π) or C / (2π)
Diameter d	2r	C / π
Circumference C	2πr ≈ 6.2832r	π × d
Area A	πr² ≈ 3.14159r²	C² / (4π)
Arc length	(θ / 360°) × 2πr	(θ in degrees)
Sector area	(θ / 360°) × πr²	(θ in degrees)
Chord length	2r × sin(θ / 2)	(θ = central angle)
Segment area	(r² / 2)(θ_rad − sin θ)	(θ in radians)

How to Use

1
Choose your known measurement: Click one of the four buttons, Radius, Diameter, Circumference, or Area, to tell the calculator which measurement you already know.
2
Enter the value and unit: Type the number and select the appropriate unit (mm, cm, m, km, in, ft, or yd). For area input, the unit label updates automatically (e.g. cm²).
3
Add a sector angle (optional): For arc length, sector area, chord length, or chord segment area, enter the central angle in degrees (0° to 360°). Leave blank if you only need the basic circle properties.
4
Set decimal precision: Choose 2, 4, 6, or 8 decimal places for the results. For most engineering work, 4 places is sufficient; 6–8 places for scientific calculations.
5
Calculate and explore: Press Enter or click Calculate. Read the SVG diagram, stat cards, full properties table, and optional step-by-step derivation. Click Copy to copy all results to clipboard.

Example Calculation

A circular garden has an area of 50 m². Find the radius, diameter, circumference, and, for the irrigation arc, the arc length and sector area for a 120° sector.

Given: Area A = 50 m², Sector angle θ = 120°

Step 1, Find radius:

r = √(A / π) = √(50 / 3.14159) = √15.9155 = 3.9894 m

Step 2, Diameter:

d = 2r = 2 × 3.9894 = 7.9789 m

Step 3, Circumference:

C = 2πr = 2 × 3.14159 × 3.9894 = 25.066 m

Step 4, Arc length (120° sector):

Arc = (120/360) × C = (1/3) × 25.066 = 8.355 m

Step 5, Sector area (120° sector):

Sector area = (120/360) × A = (1/3) × 50 = 16.667 m²

Quick mental check, does this make sense?

A 120° sector is exactly one-third of the full circle (120/360 = 1/3). So the sector area should be exactly 50/3 ≈ 16.667 m² and the arc length exactly C/3 ≈ 8.355 m. Any answer that differs from these is a sign of a calculation error.

Understanding Circle | Area & Circumference

π (Pi), The Number That Defines Every Circle

Every circle formula involves π, the ratio of a circle's circumference to its diameter. It is the same for every circle ever drawn, no matter the size. This remarkable constancy was noticed by ancient mathematicians across cultures: Babylonians used 25/8 = 3.125, Egyptians approximated 256/81 ≈ 3.16, and Archimedes bounded π between 223/71 and 22/7 using 96-sided polygons, establishing one of the greatest intellectual achievements of classical antiquity.

›π is irrational: it cannot be expressed as a fraction of two integers. The decimal expansion never terminates and never repeats, a fact proved by Johann Lambert in 1761.
›π is transcendental: it is not a root of any polynomial with integer coefficients. This was proved by Ferdinand von Lindemann in 1882, which simultaneously proved that squaring the circle with compass and straightedge is impossible.
›π has been computed to over 100 trillion digits: despite this, just 39 decimal places of π are enough to compute the circumference of the observable universe to the width of a hydrogen atom.

Why 22/7 is only an approximation

22/7 ≈ 3.142857... differs from π ≈ 3.141592... in the third decimal place. It's useful for quick mental calculations but gives an error of about 0.04%. For engineering work, always use at least π ≈ 3.14159265. This calculator uses JavaScript's full double-precision value of π.

Why the Area Formula is A = πr², an Intuitive Proof

The area formula looks simple, but where does it come from? There are several elegant derivations, here is one of the most intuitive:

Method: Concentric ring integration

Think of the circle as nested rings, each of width dr and radius ρ.

Each ring has area ≈ circumference × thickness = 2πρ × dr

Total area = ∫₀ʳ 2πρ dρ = 2π × [ρ²/2]₀ʳ = 2π × r²/2 = πr²

Another approach: cut the circle into many thin wedge sectors, then rearrange them into an approximate rectangle. The rectangle's height approaches r and its base approaches half the circumference (πr), giving area = r × πr = πr². As the number of sectors approaches infinity, the approximation becomes exact.

Sectors, Arcs, Chords, and Segments

A circle contains several important sub-regions and measurements that arise constantly in design, engineering, and navigation:

›Sector (pie slice): the region bounded by two radii and the arc between them. Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². Used in pie charts, clock design, gear tooth layout, and sprinkler coverage planning.
›Arc: the curved boundary of a sector. A semicircle is a 180° arc. The arc length is proportional to the central angle.
›Chord: a straight line connecting two points on the circle. The longest chord is the diameter. Chord length = 2r × sin(θ/2), where θ is the central angle subtended.
›Segment: the region between a chord and the arc above it. Segment area = (r²/2)(θ_rad − sin θ), where θ is in radians. Used in calculating water volume in partially filled cylindrical tanks.
›Annulus: the ring-shaped region between two concentric circles. Annulus area = π(R² − r²). Common in washer/gasket design and pipe cross-sections.

The Circle in Coordinate Geometry

A circle centred at the origin with radius r is described by the equation:

x² + y² = r² (circle centred at origin)

(x − h)² + (y − k)² = r² (circle centred at point (h, k))

›This equation is derived directly from the Pythagorean theorem, any point (x, y) on the circle forms a right triangle with legs x and y and hypotenuse r.
›The unit circle (r = 1) is the foundation of trigonometry: every angle θ corresponds to a point (cos θ, sin θ) on the unit circle.
›Circles appear in physics as equipotential curves in uniform electric fields, as wave fronts from a point source, and as the cross-sections of all cylindrical and spherical objects.

Real-World Circle Measurements

Object	Known measurement	Derived properties
Soccer ball	Circumference 68–70 cm	Radius ≈ 10.9–11.1 cm, Area ≈ 373–388 cm²
Basketball (NBA)	Circumference 74.9 cm	Radius ≈ 11.92 cm, Diameter ≈ 23.8 cm
Earth equator	Radius 6,371 km	Circumference ≈ 40,030 km, Area ≈ 127.5 M km²
CD / DVD	Diameter 120 mm	Radius 60 mm, Area ≈ 11,310 mm²
12-inch pizza	Diameter 30.48 cm	Area ≈ 729 cm² (total slice surface)
Large Hadron Collider	Circumference ≈ 27 km	Radius ≈ 4.3 km, one of the largest circles built
Ferris wheel (standard)	Diameter ≈ 50 m	Circumference ≈ 157 m, Radius 25 m
Olympic running track	Inside circumference ≈ 400 m	Radius of curved ends ≈ 36.5 m

Inscribed, Circumscribed, and Related 3D Shapes

Related shape	Key dimension	Area / Volume
Inscribed square	Side = r√2	Area = 2r² (63.66% of circle)
Inscribed equilateral triangle	Side = r√3	Area = 3r²√3/4 (41.35% of circle)
Inscribed regular hexagon	Side = r	Area = 3r²√3/2 (82.70% of circle)
Circumscribed square	Side = 2r = d	Area = 4r² (circle is 78.54% of square)
Sphere with same radius	Surface area = 4πr²	Volume = (4/3)πr³
Cylinder with same radius (h=r)	Lateral surface = 2πr²	Volume = πr³

Frequently Asked Questions

What is the formula for the circumference of a circle?

›Circumference C = 2πr, where r is the radius.
›Equivalently: C = πd, where d is the diameter.
›π ≈ 3.14159265 (irrational, the decimal never ends or repeats).
›If you know the area A instead: C = 2√(πA).
›Example: r = 7 cm → C = 2 × π × 7 ≈ 43.982 cm.

How do I find the radius from the circumference?

›Rearrange C = 2πr to get: r = C / (2π).
›Divide the circumference by 2π ≈ 6.2832.
›Example: a circle with circumference 100 cm → r = 100 / 6.2832 ≈ 15.915 cm.
›From diameter: r = d / 2 (simplest formula).
›From area: r = √(A / π).
›This calculator accepts any one of these four measurements and derives the rest.

What is the difference between circumference and area?

›Circumference is the perimeter, the length of the circle's boundary. It is a 1D measurement in linear units (m, cm, in, etc.).
›Area is the surface enclosed by the circle, a 2D measurement in square units (m², cm², in², etc.).
›C = 2πr and A = πr². Area grows much faster: doubling the radius doubles the circumference but quadruples the area.
›Practical distinction: circumference tells you how much fencing you need to border a circular garden; area tells you how much soil or turf to fill it.

What is a sector of a circle, and how do I calculate its area?

›A sector (also called a "pie slice") is the region between two radii and the arc connecting them.
›Sector area = (θ / 360°) × πr², where θ is the central angle in degrees.
›Equivalently: sector area = (θ_rad / 2) × r², where θ_rad is the angle in radians.
›Example: a 90° sector of a circle with r = 10 m has area = (90/360) × π × 100 = 25π ≈ 78.54 m².
›The sector's arc length = (θ / 360°) × 2πr.
›A semicircle is a 180° sector with area = πr² / 2.

How do I find the area of a circle if I only know the circumference?

›First find the radius from the circumference: r = C / (2π).
›Then compute the area: A = πr² = π × (C / (2π))² = C² / (4π).
›Direct formula: A = C² / (4π) ≈ C² / 12.566.
›Example: C = 40 m → A = 40² / (4π) = 1600 / 12.566 ≈ 127.32 m².
›This two-step approach avoids intermediate rounding: derive r precisely first, then compute A.

What is the chord of a circle?

›A chord is a straight line segment connecting any two points on a circle.
›The longest chord passes through the centre, this is the diameter.
›Chord length = 2r × sin(θ/2), where θ is the central angle subtended by the chord.
›The perpendicular from the centre to a chord always bisects the chord.
›Chord segment area (region between chord and arc) = (r²/2)(θ − sin θ), where θ is in radians.
›Chords are important in gear design, structural arches, and cable bridge engineering.

Why is the circle considered the most efficient shape?

›Among all shapes with the same perimeter (circumference), the circle encloses the maximum area, this is the isoperimetric inequality.
›Mathematically: for any closed curve of perimeter L, the enclosed area A ≤ L² / (4π), with equality only for a circle.
›This efficiency is why bubbles, tree trunks, and cell membranes tend toward circular cross-sections, they minimise boundary length (and thus energy) for a given enclosed volume.
›Circular pipes move fluids most efficiently for the same cross-sectional area.
›Circular wheels minimise friction over any distance because every point on the rim contacts the ground equally.

What is the equation of a circle in coordinate geometry?

›Circle centred at the origin: x² + y² = r².
›Circle centred at point (h, k): (x − h)² + (y − k)² = r².
›This follows directly from the Pythagorean theorem: any point (x, y) on the circle is at distance r from the centre.
›Expanding (x − h)² + (y − k)² = r² gives the general form: x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, F = h² + k² − r².
›The unit circle (r = 1, centred at origin) is the foundation of trigonometry, sin θ and cos θ are defined as the y and x coordinates of the point at angle θ.

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