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Circle calculator: area, circumference and radius

Area, circumference, radius and diameter of a circle from any one of them, plus arc length, chord, sector and segment area for a central angle.

Updated Checked against 6 worked examples

Between 0° and 360°
Try
Area
cm²
Area: 78.5398 cm²
Shown to up to 4 decimal places, half-up
Radius
5cm
Diameter
10cm
Circumference
31.4159cm
Arc length
5.236cm
Chord length
5cm
Sector area
13.09cm²
Segment area
2.2647cm²

A circle of radius 5 cm has diameter 10 cm, circumference 31.4159 cm and area 78.5398 cm². A 60° arc of it is 5.236 cm long, and the chord joining its ends is 5 cm.

Circle, sector and chord

r = 5 cmd = 10 cm60°arc 5.236 cmO
How it's calculated S

  1. Radius

    r=5 cmr = 5\,\text{cm}

  2. Diameter and circumference

    d=2r=10 cm,C=2πr=2π×5=31.415927 cmd = 2r = 10\,\text{cm},\qquad C = 2\pi r = 2\pi \times 5 = 31.415927\,\text{cm}

  3. Area

    A=πr2=π×52=78.539816 cm2A = \pi r^2 = \pi \times 5^2 = 78.539816\,\text{cm}^{2}

  4. Arc length and chord

    s=rθ=5×60π180=5.235988 cm,c=2rsin⁡θ2=2×5×sin⁡30∘=5 cms = r\theta = 5 \times \frac{60\pi}{180} = 5.235988\,\text{cm},\qquad c = 2r\sin\frac{\theta}{2} = 2 \times 5 \times \sin 30^\circ = 5\,\text{cm}

  5. Sector and segment areas

    Asector=πr2θ360∘=13.089969 cm2,Asegment=r22(θ−sin⁡θ)=2.264652 cm2A_\text{sector} = \pi r^2 \frac{\theta}{360^\circ} = 13.089969\,\text{cm}^{2},\qquad A_\text{segment} = \frac{r^2}{2}\left(\theta - \sin\theta\right) = 2.264652\,\text{cm}^{2}

    The segment is the region between the chord and the arc; θ is in radians inside the bracket.

About the circle calculator

Give any one of the radius, diameter, circumference or area and the other three follow from d = 2r, C = 2πr and A = πr². For a central angle θ the calculator also returns the arc length s = rθ (θ in radians), the chord c = 2r sin(θ/2), the sector area πr² × θ/360° and the segment area r²/2 × (θ − sin θ).

The default radius of 5 cm gives a diameter of 10 cm, a circumference of 31.4159 cm and an area of 78.5398 cm². Its 60° arc is 5.236 cm long and the chord across it is exactly 5 cm, because two radii and that chord form an equilateral triangle. Starting from the circumference suits trees, pipes and columns whose diameter cannot be reached: a girth of 100 cm means a diameter of 31.831 cm.

Inputs can be in any length or area unit, including hectares and acres, and every result appears in the unit chosen under "Show results in". The central angle must be above 0° and at most 360°; at 360° the segment is the whole disc.

Worked examples

Radius 5 cm, 60° arc

I know the
Radius
Radius
5 cm
Central angle for arc and chord
60 °
Show results in
Centimetres (cm)
Area
78.539816 cm²
Circumference
31.415927 cm
Diameter
10 cm
Arc length
5.235988 cm
Chord length
5 cm
Sector area
13.089969 cm²

Checked against: Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360

Circumference 100 cm

I know the
Circumference
Circumference
100 cm
Central angle for arc and chord
60 °
Show results in
Centimetres (cm)
Radius
15.915494 cm
Area
795.774715 cm²
Diameter
31.830989 cm

Checked against: Python 3.8 math: 100/(2*pi), 100**2/(4*pi)

Area 1 m², results in metres

I know the
Area
Area
1 m²
Central angle for arc and chord
60 °
Show results in
Metres (m)
Radius
0.56419 m
Circumference
3.544908 m

Checked against: Python 3.8 math: sqrt(1/pi), 2*sqrt(pi)

Diameter 12 in, quarter arc

I know the
Diameter
Diameter
12 in
Central angle for arc and chord
90 °
Show results in
Inches (in)
Chord length
8.485281 in
Sector area
28.274334 in²
Segment area
10.274334 in²
Arc length
9.424778 in

Checked against: Python 3.8 math: 12*sin(pi/4), 36*pi/4, 36*pi/4 − 18, 6*pi/2

Questions

How do you find the area of a circle from the diameter?

Use A = πd²/4, which is πr² with the radius written as half the diameter. A 10 cm diameter gives 25π ≈ 78.54 cm², and a 12 in pizza covers 36π ≈ 113.10 in². Area grows with the square of the diameter, so one 16 in pizza (201.06 in²) has more area than two 11 in pizzas together (190.07 in²).

How do you find the radius from the circumference?

Divide the circumference by 2π: r = C/(2π). A tree with a girth of 100 cm has a radius of 15.915 cm and a diameter of 31.831 cm (C/π). Foresters' diameter tapes do this division for you: their scale is graduated in units of π, so wrapping the tape round the trunk reads the diameter directly.

What is the difference between a sector and a segment of a circle?

A sector is the pie slice between two radii and the arc; a segment is the region between the arc and the chord joining its ends. The segment is the sector minus the triangle formed by the two radii and the chord. For a 90° slice of a circle with a 6 in radius, the sector is 9π ≈ 28.274 in², the triangle 18 in², and the segment 10.274 in².

How do you calculate arc length?

Multiply the radius by the central angle in radians: s = rθ. With the angle in degrees, use s = 2πr × θ/360. A 60° arc on a 5 cm radius is 5 × π/3 ≈ 5.236 cm, one sixth of the 31.416 cm circumference. Putting degrees straight into s = rθ gives an answer 57.3 times too large, the number of degrees in one radian.

Is 3.14 accurate enough for pi?

For estimates, yes. 3.14 is 0.05% below π, so a 5 cm radius gives an area of 78.5 cm² instead of 78.54 cm²; the fraction 22/7 is 0.04% too high. The absolute error grows with size: for a radius of 100 m, 3.14 gives 31,400 m², which is 15.9 m² short of the true 31,415.9 m².

How accurate is the circle calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “Radius 5 cm, 60° arc” is checked against Python 3.8 math: pi*5**2, 2*pi*5, 5*radians(60), 2*5*sin(radians(30)) = 5 (equilateral triangle), pi*25*60/360.

Where does the method come from?

Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld; NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π.

About this calculator

A=πr2,C=2πr,s=rθc=2rsin⁡θ2,Asegment=r22(θ−sin⁡θ)\begin{gathered} A = \pi r^2,\quad C = 2\pi r,\quad s = r\theta \\ c = 2r\sin\tfrac{\theta}{2},\quad A_\text{segment} = \tfrac{r^2}{2}(\theta - \sin\theta) \end{gathered}

Sources

  1. Weisstein, E. W. “Circle”, “Circular Segment” — MathWorld
  2. NIST Digital Library of Mathematical Functions §3.12 — mathematical constant π

Checked against references

6 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.

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