Calculate radius, diameter, circumference, area, arc length and other circle properties instantly with formulas, diagrams and step-by-step explanations.
Every circle measurement follows from just one. If you know the radius, the diameter is d = 2r, the circumference is C = 2πr and the area is A = πr². Working backwards, r = d/2, r = C/(2π) and r = √(A/π). Enter any single value below — radius, diameter, circumference or area — and the calculator fills in the rest, then adds arc length, sector area, chord and segment for any central angle you choose.
Type into any one field and the other three update automatically. Decimals are fine — nothing is rewritten while you type.
Arc, Sector, Chord & Segment
| Property | Value |
|---|
| Unit | Area |
|---|
Need areas for other shapes? Use the area calculator.
A circle is the set of all points in a plane that sit exactly the same distance from a fixed central point. That single defining property — constant distance from the centre — is what produces every formula on this page. The fixed distance is the radius, and the fixed point is the centre.
Strictly speaking the circle is the curve itself; the flat region enclosed by it is called a disc. In everyday use "circle" covers both, which is why we talk about the area of a circle even though area belongs to the region inside. Circles are perfectly symmetrical about every line through the centre, and of all shapes with a given perimeter a circle encloses the largest possible area — the reason bubbles, planets and pressure vessels are round.
| Term | Definition |
|---|---|
| Centre | The fixed point every point on the circle is equidistant from |
| Radius (r) | Distance from the centre to the edge |
| Diameter (d) | Distance straight across through the centre; always 2r |
| Circumference (C) | The distance all the way around the circle |
| Area (A) | The surface enclosed by the circle, in square units |
| Chord | A straight line joining two points on the circle; the diameter is the longest chord |
| Arc | Part of the circumference between two points |
| Sector | A "pizza slice" bounded by two radii and an arc |
| Segment | The region between a chord and its arc |
| Tangent | A line touching the circle at exactly one point, always perpendicular to the radius there |
| Secant | A line cutting the circle at two points |
| Central angle (θ) | The angle at the centre between two radii |
| Sagitta | The height of an arc above its chord |
| Find | Formula | From |
|---|---|---|
| Diameter | d = 2r | Radius |
| Radius | r = d/2, r = C/(2π), r = √(A/π) | Diameter, circumference or area |
| Circumference | C = 2πr = πd | Radius or diameter |
| Area | A = πr² = πd²/4 = C²/(4π) | Radius, diameter or circumference |
| Arc length | L = rθ = (θ°/360) × 2πr | Radius and central angle |
| Sector area | A = ½r²θ = (θ°/360) × πr² | Radius and central angle |
| Chord length | c = 2r·sin(θ/2) | Radius and central angle |
| Segment area | A = ½r²(θ − sin θ) | Radius and central angle |
| Sagitta (arc height) | h = r(1 − cos(θ/2)) | Radius and central angle |
In every formula containing θ on its own, the angle must be in radians; the versions dividing by 360 take degrees. Converting is simple: radians = degrees × π/180.
The radius is the distance from the centre to any point on the circle, and it is the value every other property is built from. If you have been given something else, convert first: r = d/2, r = C/(2π), or r = √(A/π). Measuring a physical circle is often easier across the full width, so measure the diameter and halve it.
d = 2r. The diameter passes through the centre and is the longest possible chord. Pipes, tanks, wheels and drill bits are all specified by diameter rather than radius, which is why substituting a diameter into A = πr² is such a common and costly slip — it makes the answer four times too large.
C = 2πr = πd. This is the perimeter of the circle: the distance a wheel travels in one full revolution, or the length of edging needed around a round garden bed. A radius of 7 gives C = 2π × 7 ≈ 43.98. The ratio C/d equals π for every circle in existence, which is precisely how π is defined.
A = πr². This is the surface enclosed, always in square units. A radius of 7 gives π × 49 ≈ 153.94 square units. Two useful alternative forms save a conversion step: A = πd²/4 when you know the diameter, and A = C²/(4π) when you know the circumference. Note that area grows with the square of the radius, so doubling the radius quadruples the area.
L = rθ with θ in radians, or L = (θ°/360) × 2πr in degrees. An arc is simply the fraction of the circumference covered by the central angle. A 90° arc is a quarter of the way around, so for r = 7 the arc measures 0.25 × 43.98 ≈ 11.00.
A = ½r²θ or (θ°/360) × πr². A sector is the pie slice between two radii. Using the same r = 7 circle, a 90° sector is a quarter of the area: 0.25 × 153.94 ≈ 38.48. Sectors turn up in pie charts, fan-shaped garden beds, sprinkler coverage and gear design.
c = 2r·sin(θ/2). A chord is the straight line between the two ends of an arc. With r = 10 and θ = 60°, c = 20 × sin 30° = 10 — the chord equals the radius, which is why six equilateral triangles fit exactly inside a circle. At θ = 180° the chord becomes the diameter, its maximum possible length.
A = ½r²(θ − sin θ). A segment is the region caught between a chord and its arc — the sector minus the triangle formed by the two radii and the chord. This is the formula engineers use for the cross-section of liquid in a partly filled round tank or pipe.
The central angle sits at the centre between two radii and determines the size of the arc, sector, chord and segment. Rearranging gives it back from what you know: θ = L/r radians from an arc length, or θ° = (sector area ÷ circle area) × 360. A full turn is 360° or 2π radians.
π is the ratio of any circle's circumference to its diameter, approximately 3.14159265358979. It is irrational, so its decimal expansion never terminates or repeats, and transcendental, meaning it is not the root of any polynomial with rational coefficients. Archimedes bounded it between 3¹⁰⁄₇₁ and 3¹⁄₇ using inscribed and circumscribed polygons around 250 BCE. For practical work 3.14159 is ample: even NASA uses only about 15 decimal places for interplanetary navigation.
| Field | Use | Example |
|---|---|---|
| Construction | Circular slabs, columns, water tanks, manholes | A 3 m diameter tank base is π × 1.5² ≈ 7.07 m² |
| Plumbing | Pipe cross-sections and flow capacity | A 100 mm pipe has a bore area of π × 50² ≈ 7,854 mm² |
| Engineering | Shafts, bearings, gears, pressure vessels | Stress calculations use the cross-sectional area πr² |
| Landscaping | Round beds, ponds, sprinkler coverage | A sprinkler throwing 6 m covers π × 36 ≈ 113 m² |
| Wheels and machinery | Distance travelled per revolution | A 700 mm wheel covers π × 0.7 ≈ 2.2 m per turn |
| Sport | Field circles, running track curves | Athletics lane staggers come from arc-length differences |
| Design | Logos, tables, pie charts, clock faces | Each pie chart slice is a sector with θ = 3.6° per percent |
| Education | Geometry, trigonometry and measurement | Australian Curriculum measurement and space strands |
A circle is two-dimensional; a sphere is its three-dimensional counterpart, the set of points a fixed distance from a centre in space. The formulas are related but distinct, and mixing them is a frequent exam error.
| Property | Circle (2D) | Sphere (3D) |
|---|---|---|
| Boundary measure | Circumference C = 2πr | Surface area A = 4πr² |
| Interior measure | Area A = πr² | Volume V = ⁴⁄₃πr³ |
| Units | Length and square units | Square and cubic units |
| Cross-section | — | Every cross-section through the centre is a circle of radius r |
A sphere's surface area is exactly four times the area of its own great circle, a result Archimedes proved and reportedly asked to have carved on his tomb. For solids, use the surface area calculator and the volume calculator.
Circle geometry is older than written proof. Egyptian scribes were calculating circular areas by about 1800 BCE: the Rhind papyrus instructs the reader to take eight ninths of the diameter and square it, which is equivalent to using π ≈ 3.1605 — remarkably close for a purely practical rule with no theory behind it. Babylonian tablets from a similar period used 3, and sometimes 3⅛.
The Greeks turned those recipes into mathematics. Around 300 BCE Euclid's Elements defined the circle from first principles — all points equidistant from a centre — and proved its properties by deduction rather than measurement, including the theorems about chords, tangents and inscribed angles still taught today. Half a century later Archimedes trapped π between inscribed and circumscribed 96-sided polygons, establishing that it lies between 3¹⁰⁄₇₁ and 3¹⁄₇, and proved that a sphere's surface area is exactly four times the area of its great circle. He reportedly asked for the sphere-and-cylinder diagram to be carved on his tomb.
Modern engineering inherited all of it. Every shaft, bearing, pipe, wheel, gear and pressure vessel is designed with the same formulas, now standardised through SI units so that a radius measured in Melbourne means precisely what it does anywhere else.
| Mistake | Why it goes wrong | How to avoid it |
|---|---|---|
| Using the diameter as the radius | πd² is four times the true area | Halve the diameter before squaring |
| Confusing circumference with area | They answer different questions and use different units | Circumference is a length; area is squared |
| Squaring 2πr instead of r | A = πr², not (2πr)² | Only the radius is squared |
| Using degrees where radians are required | L = rθ and ½r²θ both need radians | Multiply degrees by π/180, or use the /360 forms |
| Rounding π too early | Using 3.14 introduces about 0.05% error, which compounds | Keep full precision and round only at the end |
| Mixing units | Millimetres with metres gives a meaningless result | Convert everything to one unit first |
| Confusing a sector with a segment | A sector includes the triangle; a segment excludes it | Segment = sector − triangle |
| Applying circle formulas to a sphere | Surface area is 4πr², not πr² | Check whether the object is flat or solid |
| Assuming doubling the radius doubles the area | Area scales with the square of the radius | Double the radius and the area quadruples |
Lengths convert with a simple factor; areas convert with the square of that factor. Angles convert between degrees and radians.
| Conversion | Factor |
|---|---|
| Radius ↔ diameter | d = 2r | r = d/2 |
| Circumference ↔ radius | C = 2πr | r = C/(2π) |
| Area ↔ radius | A = πr² | r = √(A/π) |
| Degrees → radians | × π/180 (so 180° = π rad, 90° = π/2) |
| Radians → degrees | × 180/π |
| 1 m ↔ mm, cm | = 1,000 mm = 100 cm |
| 1 m² | = 10,000 cm² = 1,000,000 mm² ≈ 10.7639 ft² |
| 1 in ↔ mm | = 25.4 mm exactly |
d = 2rr = d/2 = C/(2π) = √(A/π)C = 2πr = πdA = πr² = πd²/4 = C²/(4π)L = rθ = (θ°/360)·2πrA = ½r²θ = (θ°/360)·πr²c = 2r·sin(θ/2)A = ½r²(θ − sinθ)h = r(1 − cos(θ/2))× π/180 and × 180/πsurface = 4πr², volume = ⁴⁄₃πr³3.14159265358979
1) d = 2 × 6 = 12 cm 2) C = 2π × 4 ≈ 25.13 m 3) A = π × 9 ≈ 28.27 cm² 4) r = 25/2 = 12.5 mm 5) Exactly π ≈ 3.14159, for every circle
1) r = 31.4/(2π) ≈ 4.997 m; A = πr² ≈ 78.46 m² 2) r = √(78.54/π) ≈ 5.00 cm, so d ≈ 10 cm 3) L = (120/360) × 2π × 9 = 6π ≈ 18.85 cm 4) A = (120/360) × π × 81 = 27π ≈ 84.82 cm² 5) c = 2 × 10 × sin 30° = 10 cm 6) ½ × 49 × (π/2 − 1) ≈ 13.98 cm² 7) C = π × 0.7 ≈ 2.199 m per revolution, so about 219.9 m 8) It becomes nine times larger, because area scales with r²
What is a circle?
A circle is the set of all points in a plane that lie exactly the same distance from a fixed central point. That distance is the radius and the fixed point is the centre. Strictly the circle is the curve itself, while the region it encloses is a disc, though everyday usage covers both. Of all shapes with a given perimeter, a circle encloses the largest possible area.
How do you calculate the area of a circle?
Use A = πr², where r is the radius. A circle with a radius of 7 has an area of π × 49 ≈ 153.94 square units. If you know the diameter instead, either halve it first or use A = πd²/4. From the circumference, A = C²/(4π). Area is always in square units.
How do you calculate the circumference of a circle?
Use C = 2πr, or equivalently C = πd. A radius of 7 gives C = 2π × 7 ≈ 43.98, and a diameter of 14 gives the same answer. Circumference is the distance all the way around the circle, so it is a length rather than an area — it is what a wheel travels in one revolution.
What is the formula for a circle?
The core set is d = 2r, C = 2πr = πd and A = πr². Rearranged for the radius they give r = d/2, r = C/(2π) and r = √(A/π). For parts of a circle, arc length is L = rθ, sector area is ½r²θ and chord length is 2r·sin(θ/2), with θ in radians. In coordinate geometry the equation of a circle centred at (h, k) is (x − h)² + (y − k)² = r².
How do I find the radius?
It depends on what you already know. From the diameter, r = d/2. From the circumference, r = C/(2π) — so a circumference of 50 gives r ≈ 7.958. From the area, r = √(A/π) — so an area of 200 gives r ≈ 7.979. For a physical object, the most accurate approach is to measure the diameter across the widest point and halve it, since locating the exact centre by eye is unreliable.
How do I find the diameter?
Double the radius: d = 2r. From the circumference, d = C/π. From the area, d = 2√(A/π). Pipes, tanks, wheels, fittings and drill bits are all specified by diameter rather than radius, which is why converting carefully matters.
How do I calculate a circle from its circumference?
Divide by 2π to get the radius: r = C/(2π). With C = 50, r = 50/6.28319 ≈ 7.9577, so the diameter is about 15.9155 and the area is πr² ≈ 198.94. You can also go straight to area with A = C²/(4π), which avoids the intermediate rounding step.
How do I calculate a circle from its area?
Take the square root of the area divided by π: r = √(A/π). With A = 200, r = √63.662 ≈ 7.9788, giving a diameter of about 15.958 and a circumference of about 50.13. This is the calculation you need when a required floor or coverage area is known and you have to work out how big to make the circle.
What is π?
π is the ratio of any circle's circumference to its diameter, approximately 3.14159265358979. It is the same for every circle regardless of size, which is what makes it a constant. π is irrational, so its decimals never terminate or repeat, and transcendental, meaning it is not a root of any polynomial with rational coefficients. Archimedes bounded it between 3¹⁰⁄₇₁ and 3¹⁄₇ around 250 BCE; 3.14159 is more than enough precision for everyday work.
What is the difference between radius and diameter?
The radius runs from the centre to the edge; the diameter runs all the way across through the centre. The diameter is exactly twice the radius, so d = 2r and r = d/2. Because A = πr² squares the radius, substituting a diameter by mistake makes the area four times too large — the single most common circle error.
How do you calculate arc length?
Multiply the radius by the central angle in radians: L = rθ. In degrees, L = (θ°/360) × 2πr, since an arc is simply that fraction of the full circumference. A 90° arc in a circle of radius 7 is a quarter of 43.98, so about 11.00. A 120° arc with radius 9 gives (120/360) × 2π × 9 = 6π ≈ 18.85.
What is a chord?
A chord is a straight line joining any two points on a circle. Its length is c = 2r·sin(θ/2), where θ is the central angle between the two points. The diameter is the longest possible chord, occurring when θ = 180°. Interestingly, at θ = 60° the chord equals the radius exactly, which is why six equilateral triangles fit perfectly inside a circle.
What is a sector of a circle?
A sector is the pie-slice region bounded by two radii and the arc between them. Its area is ½r²θ in radians, or (θ°/360) × πr² in degrees. A 90° sector is a quarter of the circle, so for radius 7 it is about 38.48 square units. Pie chart slices are sectors, with each percentage point corresponding to 3.6°.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, like a pizza slice. A segment is bounded by a chord and its arc — the sector with the central triangle removed. So segment area = sector area − triangle area = ½r²(θ − sin θ). Segments matter in engineering for calculating the cross-section of liquid in a partly filled round tank or pipe.
What is a tangent?
A tangent is a straight line that touches a circle at exactly one point without crossing it. The key property is that a tangent is always perpendicular to the radius drawn to the point of contact, which makes it the basis of many geometry proofs. A line crossing the circle at two points is called a secant instead. From an external point, the two tangents drawn to a circle are always equal in length.
How do I calculate the central angle?
From an arc length, θ = L/r radians, which you convert to degrees by multiplying by 180/π. From a sector area, θ = 2A/r² radians, or as a proportion, θ° = (sector area ÷ total circle area) × 360. From a chord, θ = 2·arcsin(c/(2r)). A full circle is 360° or 2π radians.
What units are used for circles?
Radius, diameter, circumference, chord and arc length are lengths, so they use millimetres, centimetres, metres or the imperial equivalents. Area and sector area use square units such as mm², cm² and m². Angles use degrees or radians. Australia uses the metric system, so millimetres are standard in construction and engineering drawings and metres for larger dimensions.
How do builders use circle calculations?
Constantly — circular slabs and footings, concrete columns, water tanks, manholes, culverts, post holes and curved brickwork all need circle geometry. A 3 m diameter tank base is π × 1.5² ≈ 7.07 m² of concrete per unit thickness. Curved walls and driveways use arc lengths, and pipe capacity comes from the cross-sectional area πr². Setting out a circle on site is done with a string line as an improvised compass.
How do engineers calculate circles?
Cross-sectional area πr² feeds directly into stress, flow and heat-transfer calculations, so shafts, bolts, pipes and cables are all analysed through it. Because area scales with the square of the radius, a modest increase in a shaft's diameter produces a large gain in strength. Rotating machinery uses circumference to convert between rotational and linear speed, and segment area gives the liquid level in a cylindrical tank.
Can one circle measurement determine all others?
Yes. A circle has only one independent dimension, so any single measurement — radius, diameter, circumference or area — fixes every other property exactly. That is what this calculator does: enter one value and the remaining three are derived immediately, along with arc, sector, chord and segment figures once you supply a central angle.
What is the difference between a circle and a sphere?
A circle is a flat two-dimensional shape; a sphere is its three-dimensional equivalent. A circle has circumference 2πr and area πr². A sphere has surface area 4πr² and volume ⁴⁄₃πr³. A sphere's surface area is exactly four times the area of its own great circle — a result Archimedes proved and reportedly wanted engraved on his tomb.
Why does area use r² but circumference use r?
Because area measures two dimensions and circumference measures one. Multiplying two lengths gives square units, so the radius appears twice; circumference is a single distance, so it appears once. This is why doubling the radius doubles the circumference but quadruples the area, and tripling it makes the area nine times larger.
What are common mistakes when calculating circles?
Substituting the diameter where the radius belongs, which makes the area four times too big; confusing circumference with area; squaring 2πr instead of just r; using degrees in formulas that require radians; rounding π to 3.14 too early; mixing millimetres with metres; confusing a sector with a segment; and applying circle formulas to spheres, where the surface area is 4πr² rather than πr².
Does this circle calculator show the formula and steps?
Yes. It displays which formula was used to derive the radius from your input, then the working for the diameter, circumference and area, plus the full arc, sector, chord, segment and sagitta calculations when you enter a central angle. It also draws a scale diagram showing the radius, diameter, chord and shaded sector, converts the area into other units, and lets you copy everything with one tap.
A one-page reference with every circle formula, the degree–radian conversions and the sphere equivalents. Use the button to print it or save it as a PDF — the rest of the page is hidden from the printout.
MegaCalcOnline.com · Circle properties, arcs, sectors, chords and segments
| Find | Formula | Notes |
|---|---|---|
| Diameter | d = 2r | Longest chord in the circle |
| Radius from d | r = d / 2 | Measure across the widest point |
| Radius from C | r = C / (2π) | |
| Radius from A | r = √(A / π) | |
| Circumference | C = 2πr = πd | A length, not an area |
| Area | A = πr² = πd²/4 = C²/(4π) | Square units |
| Arc length | L = rθ = (θ°/360) × 2πr | θ alone must be in radians |
| Sector area | A = ½r²θ = (θ°/360) × πr² | Two radii plus an arc |
| Chord length | c = 2r·sin(θ/2) | Equals r when θ = 60° |
| Segment area | A = ½r²(θ − sinθ) | Sector minus the triangle |
| Sagitta (arc height) | h = r(1 − cos(θ/2)) | Height of the arc above its chord |
| Central angle | θ = L/r = 2A/r² = 2·arcsin(c/2r) | Radians |
| Semicircle | A = ½πr², P = πr + 2r | Perimeter includes the diameter |
| Equation of a circle | (x − h)² + (y − k)² = r² | Centre (h, k) |
| Conversions & constants | Value |
|---|---|
| π | ≈ 3.14159265358979 |
| Degrees → radians | × π/180 (180° = π, 90° = π/2, 60° = π/3) |
| Radians → degrees | × 180/π |
| Full circle | 360° = 2π radians |
| Scaling rule | double r → double C, quadruple A |
| Sphere (3D) | surface = 4πr², volume = ⁴⁄₃πr³ |
| 1 m² | = 10,000 cm² = 1,000,000 mm² ≈ 10.7639 ft² |
Educational use only. Where θ appears without /360, it must be in radians.