Skip to content
Math & Statisticsgeometry

Circle Calculator

Every circle property from any one of four measurements, including sector, arc and segment. The circumference is exactly the derivative of the area with respect to radius, which is why growing a circle by a thin ring adds area equal to the circumference times the thickness.

Also called: circle area calculator, circumference calculator.

degrees
Area
78.539816

Area 78.539816, circumference 31.42, radius 5 and diameter 10. Enter a sector angle to get the sector, arc, chord and segment as well. A square that fits inside is 7.07 on a side, and one that contains it is 10.

Radius
5
Diameter
10
Circumference
31.42
Sector area
0
Arc length
0
Chord across the sector
0
Segment area
0
Largest square that fits inside
7.07
Smallest square that contains it
10
Volume if spun into a sphere
523.6
On the sector
Enter a sector angle to get the sector, arc, chord and segment as well.
How this is calculated
Method and background

How this is calculated

Whichever measurement you have solves back to the radius and everything else follows. The relationship worth noticing is that differentiating the area formula gives the circumference: adding a ring of thickness dr adds area 2 pi r dr, the circumference times the thickness. The sector figures cover the cases that come up in practice, from pie charts to cutting pipe, and the inscribed and circumscribed squares are what tells a fabricator what stock a disc comes out of.

the circumference is the derivative of the area with respect to radius, which is why the formulas look related
r
Radius
theta
Sector angle in degrees

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a radius of 5

I know the
Radius
Value
5
Sector angle, if you need one
0 degrees

Area78.539816

pi times 25; the inscribed square side is r root 2

Open this example

a 90 degree sector is a quarter

I know the
Radius
Value
5
Sector angle, if you need one
90 degrees

Area78.539816

boundary: exactly a quarter of the area and circumference

Open this example

Method and limits

What it assumes

  • A perfect circle, with all outputs in the same unit as the input.

What it deliberately does not model

  • The segment area is the sector less the triangle, which goes negative above 180 degrees if computed naively; it is handled here.
  • Units are whatever you enter, consistently: a radius in metres gives an area in square metres.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Why is the circumference formula the derivative of the area formula?
Because growing the radius slightly adds a thin ring whose area is the circumference times the thickness. The same relationship holds between a sphere volume and its surface area.
What is the difference between a sector and a segment?
A sector is the pie slice bounded by two radii and the arc. A segment is the smaller piece cut off by the chord alone, so it is the sector less the triangle.