Area of a Circle Calculator

Area of a circle is A = π × r². Pi (π) is approximately 3.14159 — the ratio of any circle's circumference to its diameter, a constant of geometry. Squaring the radius is what makes circle area grow so quickly with size: double the radius and the area increases by a factor of 4 (2²); triple the radius and area grows 9×. This is why a 16-inch pizza has roughly twice the pizza of two 12-inch pizzas, despite costing about the same money.

The (4r²) bounding-square comparison is a useful sanity check. A circle inscribed in a square of side 2r has area πr² ≈ 0.785 × (2r)². That ratio of about 78.5% is fixed for every circle and gives you a quick mental check when something looks off.

Pizza is the textbook example. A 12-inch pizza has radius 6 in and area π × 36 ≈ 113 in². A 16-inch pizza has area π × 64 ≈ 201 in² — 78% more pizza than the 12, not the 33% that the diameter difference suggests. At $14 for the 12 and $19 for the 16, the per-square-inch prices are $0.124 and $0.094 respectively. The 16 is the obvious value if you can finish it.

A center-pivot irrigation system has a 400-meter arm and waters a full circle. Area = π × 400² ≈ 502,655 m² ≈ 124 acres. At a typical Great Plains corn yield of 230 bushels per acre, that single pivot grows about 28,500 bushels per season. Doubling the arm length to 800 m would quadruple the watered area to 248 acres — which is also why pivots are usually limited to a quarter-section (160 acres) on a typical homestead grid.

From diameter: divide by 2, then apply A = π × r². Or use A = π × d² / 4 directly. A 10-inch pan has area π × 100 / 4 ≈ 78.54 in².

From circumference: solve r = C / (2π), then A = π × r². Or use the shortcut A = C² / (4π). A circle with circumference 31.4 has radius about 5 and area about 78.5.

Sectors and segments. The area of a pie-slice (sector) with central angle θ in degrees is A = π × r² × θ / 360. A segment (the area between a chord and the arc) is the sector area minus the triangle formed by the chord and the two radii.

Annulus (ring). The area of a flat ring between two concentric circles is π × (R² − r²). A washer with outer radius 1 in and inner radius 0.5 in has area π × (1 − 0.25) ≈ 2.36 in².