Circle Calculator

A circle calculator computes the area, circumference, diameter, radius, and arc length of a circle from any single input value. It is used by students studying geometry, architects, engineers, and anyone who needs to work with circular shapes in design, construction, or mathematics.

How to Use the Circle Calculator

1. Select which measurement you already know: radius, diameter, area, or circumference.
2. Enter the value and choose your unit (millimetres, centimetres, metres, inches, or feet).
3. Click Calculate to see all other measurements of the circle computed instantly.
4. For arc and sector calculations, enter the central angle in degrees or radians.
5. View the diagram showing the relationship between radius, diameter, and circumference.

The Formula

All circle calculations use the mathematical constant pi (pi = 3.14159...).

Circumference: C = 2 x pi x r = pi x d Area: A = pi x r^2 Diameter: d = 2r Radius from circumference: r = C / (2 x pi) Radius from area: r = sqrt(A / pi)

For a sector with central angle theta (in degrees): Arc length: L = (theta / 360) x 2 x pi x r Sector area: A_sector = (theta / 360) x pi x r^2

Real-World Example

A circular garden has a diameter of 8 metres. Find its area, circumference, and the length of a 90-degree arc.

Radius: r = 8 / 2 = 4 metres

Circumference: C = 2 x pi x 4 = 8 x pi = 25.13 metres

Area: A = pi x 4^2 = 16 x pi = 50.27 square metres

Arc length for 90°: L = (90/360) x 25.13 = 0.25 x 25.13 = 6.28 metres

This tells the gardener that the circular lawn has a boundary of about 25 metres (useful for fencing) and covers just over 50 square metres (useful for turf calculations).

Circles in Architecture and Engineering

Circles are among the most structurally efficient shapes, which is why they appear everywhere in construction and design. Cylindrical columns and circular arches distribute weight evenly, making them exceptionally strong. Domes (essentially half-spheres) enclose the maximum volume for a given surface area. Circular pipes are used in plumbing and infrastructure because a circle has the smallest perimeter for a given cross-sectional area, minimising material cost while maximising flow. In mechanical engineering, gears, wheels, bearings, and flywheels are all circular because rotation requires symmetric mass distribution. Track design for athletics and motorsport uses circular curves and the related geometry of tangent lines. In optics, circular lenses focus light through the symmetric geometry of their curved surfaces. Knowing how to calculate circle dimensions is a basic but constantly applied skill in technical work.

Frequently Asked Questions

What is pi and why is it important? Pi (pi) is the ratio of a circle's circumference to its diameter: C/d = pi, for every circle regardless of size. It is an irrational number, meaning its decimal expansion continues without repeating: 3.14159265... Pi appears not only in circle geometry but across mathematics, physics, probability, and engineering, making it one of the most important constants in all of science.

How do I calculate the area of a semicircle? A semicircle is half a circle. Its area is (pi x r^2) / 2. Its total perimeter is the curved part (pi x r) plus the straight diameter (2r), giving pir + 2r. For example, a semicircle with radius 5 cm has area (pi x 25) / 2 = 39.27 cm^2 and perimeter = 5pi + 10 = 25.71 cm.

What is the difference between arc length and chord length? Arc length is the distance along the curved edge of a circle between two points, measured along the circumference. Chord length is the straight-line distance between those same two points across the interior of the circle. same central angle, arc length is always greater than chord length (they are equal only in the limiting case of an infinitely small angle where both approach zero).

Can I calculate a circle's measurements from just its area? Yes. From the area A, the radius is r = sqrt(A / pi). From the radius you can find everything else. For example, if a circular field has an area of 1 hectare (10,000 square metres), then r = sqrt(10,000 / pi) = sqrt(3,183) = 56.42 metres, and the circumference is 2 x pi x 56.42 = 354.5 metres.

Also try these free tools:

• Area Calculator
• Polygon Calculator
• Triangle Calculator

An input and output table for six radii

The formulas scale in a way a single example conceals, and a short table shows it. Every figure below is in metres, to four decimal places, and each row follows from the radius alone.

Three patterns sit in those columns. Doubling the radius doubles the circumference and quadruples the area, because the circumference carries the radius to the first power and the area carries it to the second. A radius of 1 gives a circumference of 6.2832 and an area of 3.1416, which is the constant itself expressed twice, once as 2 pi and once as pi. A radius of 50 gives a circumference of 314.1593, which is 100 pi, and an area of 7853.9816, which is 2500 pi. The area grows by the square of the radius while the boundary grows in step with it, and that gap is why a large circular field needs only three times the fencing of one a third the size while its turf requirement runs to nine times.

Half the radius times the circumference

One identity ties the area and the circumference together and makes a useful check on any circle calculation. The area equals half the radius multiplied by the circumference.

• Radius 4, circumference 25.1327, half the radius 2, product 50.2655, and the area is 50.2655
• Radius 10, circumference 62.8319, half the radius 5, product 314.1593, and the area is 314.1593

Archimedes proved this relationship, and proposition one of Measurement of a Circle states it geometrically: the area of any circle is equal to a right-angled triangle whose two shorter sides about the right angle are the radius and the circumference. The triangle's area is half of radius times circumference, which is the same expression as the modern formula once the circumference is written as 2 pi r. Anyone who has both numbers to hand can check one against the other without leaving the page.

Arc and chord at the same central angle

Arc length and chord length are often treated as interchangeable, and they are close only while the angle is small. Both are measured on the radius of 4 metres from the example above, so the numbers can be compared directly.

The chord runs from one end of the arc to the other in a straight line, so it always falls short of the curve, and the gap widens with the angle. At 30 degrees the difference is about 1 percent, which is why shortening an arc to its chord is a reasonable approximation for small spans and a poor one for large. At 180 degrees the arc is a half circle and the chord is the diameter, so the ratio settles at pi divided by 2, or 1.5708, and it cannot grow past that however wide the angle becomes. The chord length itself is twice the radius multiplied by the sine of half the central angle, which is the form to use when a straight-line distance across a circle is what a project needs.

Where the rounding lands

Two decisions affect the last digits of any answer, and they are worth separating because one of them matters far more than the other.

Take the radius of 4 metres again. Carrying pi at 3.14159 gives a circumference of 25.132720 metres, against 25.132741 from the full double-precision value, a difference of 0.000021 metres, or 0.021 millimetres. The same comparison on the area gives 50.265440 square metres against 50.265482, a difference of 0.000042 square metres, or 42 square millimetres.

Now round both answers to two decimal places, as a summary would. The circumference becomes 25.13, which is up to 5 millimetres away from the value it summarises. The rounding of the answer is therefore some two hundred times larger than the truncation of pi, and on a radius of 4 metres the choice of pi is not what limits the accuracy of the result. Precision in pi costs nothing and settles nothing on its own. The precision of the input radius is what sets the ceiling, and it is worth deciding how much of it a project actually needs before an answer is quoted to four decimals.

What the circle formulas assume

The calculator treats the shape as an exact circle and the numbers as exact inputs, so every error in a result traces back to the measurement that produced it, which is the assumption to keep in view.

• Any one of the four measures determines the other three. Supplying the radius works forwards through the formulas, while supplying the area reverses them through a square root, which is why an area input must be non-negative and why a hand calculation from an area yields two radius values of equal size and opposite sign.
• Area scales with the square of a linear dimension. A radius measured 10 percent too long gives an area 21 percent too large, because 1.1 squared is 1.21. The same relationship amplifies any unit slip in the input.
• Linear measures keep the unit supplied and the area takes its square. A radius entered in centimetres returns a circumference in centimetres and an area in square centimetres, so mixing a radius in centimetres with an area in square metres is the error most likely to survive a quick check.
• Sector and arc results use the angle as a fraction of a full turn, which is the theta divided by 360 term in the formulas above. The default is degrees.
• The formulas describe a whole circle. An ellipse, a circular segment cut by a chord, and a ring between two circles all need their own relationships, and the semicircle answer in the questions above is a special case of the segment.

A note on the value of pi

Archimedes, in proposition three of Measurement of a Circle, proved that the ratio of a circle's circumference to its diameter is greater than 3 and 10/71 and less than 3 and 1/7. As decimals those bounds are 3.140845 and 3.142857, both within 0.05 percent of the modern value of 3.14159265, and he established them by inscribing and circumscribing a regular 96-sided polygon around a circle. Measured with the upper bound of 22/7, a circle of radius 4 metres would show an area of 50.2857 square metres against the 50.2655 square metres given by pi, an error of 0.04 percent. The calculator works to the full precision of a double-precision number, roughly 16 significant decimal digits, which is finer than any measurement a reader is likely to bring to it.