Circle Calculator — Area & Circumference
Calculate radius, diameter, circumference, and area of a circle from any known value.
Enter any one circle measurement to find all others
What Is the Circle Calculator — Area & Circumference?
The Circle Calculator computes all properties of a circle — radius, diameter, circumference, and area — from any one known measurement. Enter just one value (radius, diameter, circumference, or area) and the tool derives all other properties.
Formula
How to Use
Select which measurement you know from the dropdown (radius, diameter, circumference, or area). Enter its value. Click Calculate to see all other circle properties including arc length and sector area for a 90° sector.
Example Calculation
Known: radius r = 7 Diameter = 2×7 = 14 Circumference = 2×π×7 = 43.982 Area = π×7² = 153.938 sq units Known: area A = 50 r = √(50/π) = √15.915 = 3.989 C = 2×π×3.989 = 25.066 d = 2×3.989 = 7.979
Understanding Circle — Area & Circumference
The circle is one of the most perfect and studied shapes in mathematics. Its constant ratio of circumference to diameter (π) was known to ancient Babylonians, Egyptians, and Greeks, who approximated it as 3 1/7 ≈ 3.14286.
Pi (π) is an irrational and transcendental number, meaning it cannot be expressed as a fraction of two integers and is not a root of any polynomial with integer coefficients. It has been computed to over 100 trillion decimal places, yet no repeating pattern exists.
Circles appear throughout nature and science: planetary orbits (approximately circular), cross-sections of cylindrical objects, sound waves (circular wavefronts), and the unit circle (the foundation of trigonometry). The equation x² + y² = r² defines a circle centered at the origin in Cartesian coordinates.
Frequently Asked Questions
What is the relationship between circumference and diameter?
C = π × d. The ratio C/d = π (pi) for any circle — this is actually the definition of π. Pi is approximately 3.14159265..., an irrational number that never ends or repeats.
What is the difference between circumference and perimeter?
Circumference specifically refers to the perimeter (boundary length) of a circle. The word "perimeter" is the general term for any closed curve; "circumference" is the specific term for circles.
What is a sector?
A sector is a "pie slice" of a circle — the region between two radii and the arc connecting them. Its area is proportional to the central angle: A_sector = (θ/360°) × πr².
What is the area formula derived from?
The area formula A = πr² can be derived by dividing a circle into thin concentric rings (each of area 2πr·dr) and integrating from 0 to r: ∫₀ʳ 2πr dr = πr².
What is an annulus?
An annulus is the region between two concentric circles of different radii. Its area = π(R² − r²) where R is the outer radius and r is the inner radius.
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