Circle Area Calculator
Calculate the area of a circle from its radius. Formula: A = π × r². Instant result in the browser.
Formula
A = π × r² ≈ 3.14159 × r²
in which r stands for the circle radius. The area comes out in the radius unit squared (m², cm², km² and so on).
The area of a circle and the role of pi
The area enclosed by a circle of radius r comes out to A = π·r². If you measured the diameter instead, the same area is A = π·d²/4, and the perimeter (circumference) is C = 2π·r. The constant π ≈ 3.14159265... is both irrational and transcendental, which means you can't write it as a finite decimal, and it isn't the root of any polynomial with rational coefficients either.
A circle of radius 5 m has an area of π · 25 ≈ 78.54 m². The relation A = π·r² got its first rigorous proof from Archimedes around 250 BC. He used the method of exhaustion, inscribing and circumscribing regular polygons and squeezing the two bounds together until they met.
Applications and the circular sector
Circle area turns up in engine pistons (the combustion chamber cross-section), in pipe flow (volumetric flow = velocity × A), in vinyl records, lenses and telescope mirrors, and even in pizza pricing. A 35 cm pizza covers about 962 cm² while a 30 cm one covers only around 707 cm², so working out the price per cm² usually shows the bigger pie is the better deal. For a partial slice (a circular sector) with central angle θ, the area is A = (θ/360°)·π·r² in degrees, or A = (1/2)·r²·θ in radians.
FAQ
What if I only know the diameter? Reach for A = π·d²/4. A 10 m diameter gives π · 100 / 4 ≈ 78.54 m², which is exactly what a 5 m radius would give you.
Why is π irrational? Back in 1761 Lambert showed that π can't be written as a fraction of two integers. Lindemann went further in 1882 and proved π is transcendental, which is what finally settled the old question of squaring the circle with compass and straightedge: it can't be done.
How do I compute a ring (annulus)? Take the outer circle and subtract the inner one, which gives A = π·(R² − r²).
How many decimals of π should I use? For everyday engineering, 3.1416 is plenty. NASA gets by on 15 digits for interplanetary navigation, and past 40 digits there's nothing left to measure that would notice the difference.
Calculate the area of a circle
A circle's area is one of the first practical uses of pi we ever run into, from the classroom to a job around the house. The tool solves it from the radius and applies the formula A = π × r² for you.
Enter the radius (or the diameter) and the result shows up right away, with no need to recall the value of pi or square anything in your head. Good for geometry exercises, and also for working out the area of a round table, a flower bed, a pizza, any circular surface you come across.
It all runs in the browser. A quick, precise reference for a circle's area, whenever geometry shows up.
Related Tools
Circle and Ellipse Area Calculator
Computes circle area from radius and ellipse area from semi-axes a and b using the standard pi formulas.
Annulus Area
Annulus area between outer R and inner r radii: A = π(R² − r²). Useful for pipes and gears.
Cam Pitch Radius
Calculate the pitch radius of a roller-follower cam, R_p = R_b + R_r, from the base circle radius R_b (mm) and the follower roller radius R_r (mm). In ROLLER-follower cams (a bearing rolling on the cam profile, reducing friction versus flat-face or knife-edge followers), two important curves are distinguished: the real PROFILE of the cam (the physical surface the roller touches) and the PITCH curve, the locus of the roller CENTER as it follows the cam. The pitch curve is designed first (from the displacement diagram), and the real profile is obtained by 'offsetting' the roller radius from the pitch curve. The pitch radius, at the base position, is the sum of the base circle radius and the roller radius. This distinction is fundamental for a practical reason: the roller radius cannot exceed the smallest RADIUS OF CURVATURE of the pitch curve in CONCAVE regions, or the roller does not 'fit' and the cam gets an incorrect profile (undercutting), distorting the motion. So the choice of roller radius and base radius is coupled to the cam geometry. The pitch radius also enters the pressure-angle and peripheral-speed calculations. Enter the base circle radius and the roller radius.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.