How this instrument works
The area enclosed by a circle of radius r is A = πr², one of the shortest exact formulas in geometry and also one of the most telling about how area behaves. Length scales with r in direct proportion — double the radius and the circumference doubles — but area is a two-dimensional quantity, so it scales with r², the square of that same length. A circle simply attaches the smallest possible constant, π, to that square.
The formula has a proof you can see rather than just trust. Slice the disk into many thin concentric rings, cut each ring and straighten it into a short strip, then lay the strips side by side from longest to shortest. The strips assemble into something close to a triangle whose base is the full circumference, 2πr, and whose height is the radius. A triangle's area is half its base times its height, so the total works out to half of 2πr times the radius, which is πr² exactly, as the rings are taken thinner and more numerous — essentially Archimedes' own argument in Measurement of a Circle, over two thousand years old.
The r² term has a consequence people underestimate: doubling the radius does not double the area, it quadruples it, since (2r)² = 4r². A pizza with twice the radius carries four times the toppings, not two. The formula also punishes one specific slip — plugging a measured diameter into the spot meant for the radius. Since d = 2r, using d² in place of r² overstates the area by exactly a factor of four, and that is the single most common arithmetic error the Radius field is built to head off.
- Type the circle's radius into the Radius field — any unit of length is fine, provided you read the result in that same unit, squared.
- The Area field returns A = πr² immediately, carried to six decimal places.
- Working backwards? Divide Area by π and take the square root to recover the radius; the formula box below shows this rearrangement.
- Note the built-in check: a negative radius is rejected, since no circle has a radius below zero.
Worked example — a radius-5 circle
Take a circular flower bed with a radius of exactly 5 metres. Its area is A = π × 5² = π × 25 = 78.53981633974483 square metres — the figure to plan topsoil around — even though the bed's edge, its circumference, is a far smaller 2π × 5 ≈ 31.42 metres. Area and boundary length measure different things, and squaring the radius is exactly why the area pulls ahead of the perimeter as a circle grows.
Double that same bed to a radius of 10 metres and the area does not double. It becomes π × 10² ≈ 314.159265 square metres, four times the original 78.53981633974483, because (2r)² is 4r², not 2r². A small increase in radius costs disproportionately more turf, paint, or fencing, precisely because area answers a two-dimensional question while radius is only ever a single length.
Questions
What is the formula for the area of a circle?
A = πr², where r is the radius. Squaring that length reflects that area is a two-dimensional measure: scale every direction of a shape by a factor of the radius and its enclosed area scales by the square of it, circles included — the reason a radius of 5 already gives 78.53981633974483 square units in this sheet's own worked example.
Why does area use r² while circumference uses only r?
Because they measure different dimensions. Circumference is a length, so it scales in direct, linear proportion with the radius: C = 2πr. Area covers a surface, a two-dimensional quantity, so it scales with the square of the radius: A = πr². Double the radius and the boundary doubles, but the enclosed area quadruples.
Is there a calculus proof of A = πr² besides the classical one?
Yes, and it reaches the same place by a different route. Treat the disk as thin rings stacked from the centre outward; a ring at distance x has circumference 2πx and a sliver of width dx, so its own area is 2πx dx. Adding every ring from 0 out to the edge with an integral gives πx² evaluated at the edge, which is exactly πr² — matching what Archimedes found through geometry, over two thousand years earlier.
What is the most common mistake when calculating this area?
Squaring the diameter instead of the radius. Because d = 2r, using d² in place of r² overstates the area by a factor of 4 — a circle of diameter 10 has an area of π × 5² ≈ 78.54, not π × 10² ≈ 314.16. Always halve a measured diameter to a radius before it goes into A = πr².
How do I find the radius if I only know the area?
Rearrange the formula to r = √(A ⁄ π). An area of 78.53981633974483 square units, for instance, gives r = √(78.53981633974483 ⁄ π) = √25 = 5 — the exact radius this sheet's own worked example started from, recovered with no rounding beyond π itself.
Does the formula change for a sector or a ring-shaped annulus?
No — both extend it rather than replace it. A sector of angle θ, measured in radians, has area ½r²θ, which reduces to πr² exactly when θ = 2π, the full circle. An annulus, the ring between two circles sharing a centre, is the difference of two: A = π(R² − r²), for outer radius R and inner radius r.