How do I calculate the area from circumference?

Find radius r = C/(2pi), then area A = pi*r^2 = C^2/(4pi).

How do I convert degrees to radians?

Multiply degrees by pi/180. A full circle is 360 degrees = 2*pi radians.

Can I use this for semicircles?

Yes. Calculate the full circle, then use 180 degrees for the arc/sector angle to get semicircle values.

Math

Circle Calculator

Enter any one circle measurement and instantly calculate radius, diameter, circumference, and area. Plus compute arc length and sector area from an angle.

Quick Answer

All circle properties derive from the radius: Diameter = 2r, Circumference = 2πr, Area = πr². Enter any one value and this calculator computes the rest.

Enter One Known Value

Select which measurement you know and enter its value.

Radius

Results

Radius

Diameter

Circumference

31.415927

Area

78.539816

Arc Length & Sector Area

Central Angle

Arc Length

7.853982

r × θ = 5 × 1.570796 rad

Sector Area

19.634954

½r²θ = ½ × 5² × 1.570796 rad

Diagram

About This Tool

The Circle Calculator is a versatile geometry tool that computes every key property of a circle from a single known measurement. Whether you start with the radius, diameter, circumference, or area, the calculator instantly determines all four values plus arc length and sector area for any central angle you specify. It is invaluable for students learning geometry, engineers designing circular components, architects planning curved structures, and anyone who needs quick, accurate circle measurements.

The Fundamental Circle Formulas

All circle properties derive from the radius r. The diameter is simply d = 2r. The circumference (perimeter) is C = 2πr = πd, which represents the total distance around the circle. The area is A = πr², giving the total space enclosed. The constant π (pi), approximately 3.14159265, is the ratio of every circle's circumference to its diameter. These relationships have been known since ancient civilizations, with Archimedes providing one of the first rigorous approximations of π around 250 BC.

Arc Length and Sector Area

An arc is a portion of the circumference. Its length equals rθ, where θ is the central angle in radians. A sector is the "pizza slice" region between two radii and an arc. Its area equals (1/2)r²θ. If you work in degrees, convert first: radians = degrees × π/180. These formulas are critical in engineering (gear teeth, pulley systems), physics (angular displacement), and everyday applications like calculating how much paint you need for a curved wall.

Pi: The Circle Constant

Pi (π) is an irrational number, meaning its decimal representation never ends or repeats. It appears not only in circle formulas but throughout mathematics: in trigonometry, complex analysis, probability (the normal distribution), and even number theory. The value of πhas been computed to trillions of digits, though for practical calculations, 15-16 digits (as used in this calculator's floating-point arithmetic) provide more than enough precision for any engineering or scientific application.

Circles in Engineering and Design

Circles are everywhere in the engineered world. Wheels, gears, bearings, pipes, tanks, and satellite orbits all rely on circular geometry. When designing a cylindrical tank, you need the circle area for volume calculations. When machining a round part, you need the circumference to determine the length of material to wrap around it. Architects use circular geometry for domes, rotundas, and decorative elements. This calculator eliminates the need to remember formulas or reach for a scientific calculator during design work.

Converting Between Units

This calculator is unit-agnostic: enter any unit (inches, centimeters, meters, feet) and the results are in the same unit (with area in squared units). To convert units, simply multiply by the appropriate factor. For example, 1 inch = 2.54 cm, so a radius of 5 inches equals 12.7 cm. The formulas themselves are universal and do not depend on the measurement system, making this tool equally useful in imperial and metric contexts.

Frequently Asked Questions

How do I calculate the area of a circle from its circumference?

First find the radius: r = C / (2 pi). Then compute the area: A = pi * r^2. Substituting, A = C^2 / (4 pi). So if the circumference is 31.4159, the radius is about 5, and the area is about 78.54. This calculator does this conversion automatically when you select 'Circumference' as your input.

What is the difference between arc length and chord length?

Arc length is the distance along the curved portion of the circle between two points. Chord length is the straight-line distance between those same two points. Arc length is always greater than or equal to chord length (equal only for a zero-degree angle). Arc length = r * theta, while chord length = 2r * sin(theta/2).

How do I convert between degrees and radians?

Multiply degrees by pi/180 to get radians. Multiply radians by 180/pi to get degrees. A full circle is 360 degrees = 2*pi radians. A right angle is 90 degrees = pi/2 radians. This calculator accepts both units for the arc/sector angle input.

Why is pi used in circle calculations?

Pi is the ratio of a circle's circumference to its diameter. It is a fundamental mathematical constant that arises naturally from the geometry of circles. No matter how large or small a circle is, dividing its circumference by its diameter always gives pi (approximately 3.14159). This universal ratio is why pi appears in every circle formula.

Can I use this calculator for semicircles or quarter circles?

Yes. First calculate the full circle properties, then use the arc/sector section. A semicircle has a central angle of 180 degrees (pi radians), and a quarter circle has 90 degrees (pi/2 radians). The arc length and sector area will give you the perimeter and area of those partial circles respectively.

What units does this calculator use?

The calculator is unit-agnostic. Enter your value in any unit (inches, cm, meters, etc.) and the results will be in the same unit. Area will be in squared units (e.g., square inches), and all linear measurements (radius, diameter, circumference, arc length) will be in the same unit you entered.

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