Circle Calculator

A circle is a simple closed shape: the set of all points in a plane sitting the same distance from one given point, called the centre. It can also be described as the curve traced by a point that keeps a constant distance from a fixed point as it moves.

That one constraint, a fixed distance, is what makes a circle so easy to work with. Give the calculator any single measurement and the other three follow, because radius, diameter, circumference and area are all locked to each other.

Parts of a circle

• Centre (or origin): the point inside a circle that is the same distance from every point on the circle.
• Radius: the distance between the centre and any point on the circle. It is half the diameter.
• Diameter: the largest distance between any two points on a circle. By that definition it always passes through the centre, and it is twice the radius.
• Circumference: the distance around the circle, the length of one full circuit.
• Arc: part of the circumference. A major arc is more than half the circumference; a minor arc is less than half.
• Chord: a line segment joining one point of a circle to another. A chord through the centre is a diameter, which makes the diameter the longest chord there is.
• Secant: a line that cuts the circle at two points. It is a chord extended past both ends.
• Tangent: a line touching the circle at exactly one point, with the rest of the line outside. A tangent meets the radius at that point at a right angle.
• Sector: the area enclosed between two radii and the arc between them. A major sector has a central angle above 180°; a minor sector has one below 180°.

The figures below show these parts.

The constant π

Radius, diameter and circumference are tied together by π, pi, which is the ratio of a circle's circumference to its diameter. Divide any circle's circumference by its diameter, no matter the size, and you get the same number: roughly 3.14159.

π is irrational, meaning it cannot be written exactly as a fraction, and its decimal expansion neither ends nor settles into a repeating pattern. It is often approximated as $\frac{22}{7}$, which comes to 3.142857142857 and is about 0.04% high. The closer fraction $\frac{355}{113}$ matches π to six decimal places.

It is also transcendental: it is not the root of any non-zero polynomial with rational coefficients. Ancient geometers spent a great deal of effort trying to "square the circle", constructing a square with the same area as a given circle in a finite number of steps using only a compass and straightedge. That turns out to be impossible, though the proof did not arrive until 1880, when Ferdinand von Lindemann showed π is transcendental and closed the question. The work put into an impossible problem was not wasted: a great deal of mathematics is well defined today because people kept chasing it.

Circle formulas

$$D=2R$$ $$C=2\pi R$$ $$A=\pi R^2$$

where R is the radius, D the diameter, C the circumference, A the area, and π is 3.14159 to five decimal places.

Two of these are worth reading closely. Circumference grows in step with the radius: double the radius and the distance around doubles. Area does not. It grows with the square of the radius, so doubling the radius makes the area four times larger. A 12-inch pizza has an area of 113.10 square inches against 78.54 for a 10-inch, which is 44% more food for two inches of diameter.

Working from any one value

The calculator accepts whichever measurement you happen to have. These are the four routes it takes.

From the radius

Everything is direct: $D=2R$, $C=2\pi R$ and $A=\pi R^2$. For R = 2 the diameter is 4, the circumference is 4π = 12.566370614359, and the area is also 4π = 12.566370614359. The two agreeing is a coincidence of R = 2, not a rule: it happens because 2R and R² are both 4 at that radius.

From the diameter

Halve it and carry on, or use the diameter forms directly:

$$R=\frac{D}{2}$$ $$C=\pi D$$ $$A=\pi {\left(\frac{D}{2}\right)}^2$$

For D = 6 the radius is 3, the circumference is 6π = 18.849555921539, and the area is 9π = 28.274333882308.

From the circumference

$$R=\frac{C}{2\pi }$$ $$D=\frac{C}{\pi }$$ $$A=\frac{C^2}{4\pi }$$

A circumference of 10 gives a radius of 1.591549430919 and an area of 7.957747154595.

From the area

$$R=\sqrt{\frac{A}{\pi }}$$ $$D=2\sqrt{\frac{A}{\pi }}$$ $$C=2\sqrt{\pi A}$$

An area of 50 gives a radius of 3.989422804014 and a circumference of 25.06628274631. The circumference form comes from substituting the radius into C = 2πR and tidying up, since 2π√(A/π) is the same as 2√(πA).

Units, and the mistake people make with them

Radius, diameter and circumference are lengths, so they carry plain units: centimetres, inches, metres. Area is a length squared, so it comes in square units. Enter a radius in centimetres and the area is in square centimetres, not centimetres.

That squaring is also where unit conversions go wrong. There are 100 centimetres in a metre but 10,000 square centimetres in a square metre, because the conversion factor gets squared along with the length. Converting the radius first, then calculating, avoids the problem entirely.

The calculator is unit-agnostic. It works in whatever unit you type, and the answers come back in that same unit, with area squared.

Where the formulas earn their keep

Circumference answers anything that wraps or rolls. A bicycle wheel of diameter 26 inches covers 26π inches, about 81.7 inches, per revolution, which is how a bike computer turns wheel rotations into distance. Anything that needs edging, banding or a belt around it is the same calculation.

Area answers anything that covers or fills. Paint for a round table top, turf for a circular lawn, the flow capacity of a pipe. Pipe capacity is a good illustration of the squaring: a pipe of twice the diameter does not carry twice as much, it carries four times as much, because the cross-section is an area.

The sector formulas extend this to a slice. A sector's area is the full area scaled by its share of the turn, so a 90° sector holds a quarter of the circle, and its arc is a quarter of the circumference.

Common mistakes

The first is mixing up radius and diameter, which is the single most frequent error here. A measurement taken across a round object is the diameter; the formulas with R in them want half of that. Getting it wrong makes the area four times too large.

The second is using 3.14 and expecting precision. It is fine for a rough estimate and out by about 0.05%, which is nothing on a flower bed and quite a lot on a machined part. The calculator carries π at full double precision and reports twelve decimal places.

The third is treating area as though it scales with size the way length does. It does not, and the pizza comparison is the cheapest way to remember why.