Circumference 31.415927, area 78.539816. All four are derived from this one radius, so they cannot disagree.
Radius
5
given
Diameter
10
twice the radius
Circumference
31.415927
2πr — linear in r
Area
78.539816
πr² — quadratic in r
Semicircle area
39.269908
half of it
Semicircle perimeter
25.707963
half the arc PLUS the diameter
Squares around and inside
The largest square that fits inside the circle and the smallest that contains it, with the share of area the circle covers
Which
Side
Note
Square inside
7.0710678
its diagonal is the diameter
Square around
10
its side is the diameter
Coverage
78.5398%
π/4, whatever the size
Everything here comes from one radius of 5.0000000, so the four measurements cannot disagree with each other.
Circumference is LINEAR in the radius and area is QUADRATIC. Double the radius and the circumference doubles while the area quadruples — which is why a 16-inch pizza is four times an 8-inch one, not two.
The circle fills exactly π/4 of the square around it — 78.5398% — whatever its size. The corners are the other 21.46%.
A semicircle's perimeter is not half the circumference: it is half the arc PLUS the diameter closing it, which is a common slip in fencing and edging estimates.
Every figure is derived from one canonical radius, so the four measurements are always consistent with each other.
What this tool shows
Double the radius and the circumference doubles — but the area quadruples. A 16-inch pizza is four times the food of an 8-inch, not twice, and that asymmetry between a linear measure and a quadratic one is where most circle mistakes live.
Radius, diameter, circumference and area from any one
Semicircle area and perimeter
A quarter circle
The largest square that fits inside
The smallest square that contains it
What share of that square the circle covers
Any one input Always consistent Semicircle figures The π/4 identity
All four derived from one radius, so they always agree.
Updated 7 September 2026 · Works in any browser, no installation
Everything follows from the radius. d = 2r, C = 2πr, A = πr². Going backwards, r = C ÷ 2π and r = √(A ÷ π). The one to watch is that area is QUADRATIC in the radius while circumference is linear.
At a glance
Formula shown
d = 2r, C = 2\u03c0r = \u03c0d, A = \u03c0r\u00b2. Going backwards, r = C \u00f7 2\u03c0 and r = \u221a(A \u00f7 \u03c0). A semicircle\u2019s perimeter is \u03c0r + 2r, not half the circumference.
Scenario support
Sizing a round table, a pipe or a pizza; converting between a measured circumference and a radius you cannot reach; comparing two circular things by area.
Educational estimate
Planning support from the values you enter — not professional advice.
The four formulas
Diameter: d = 2r. The straight line through the centre, and always twice the radius.
Circumference: C = 2πr, equivalently πd. The distance round the edge.
Area: A = πr². The space inside.
The two worth not confusing are C = 2πr and A = πr². One has the 2 and no square; the other has the square and no 2. A mnemonic that survives: “two pie R, that’s the perimeter; pie R squared, that’s the area.”
The units are the real tell. Circumference is a length, so it comes out in centimetres. Area comes out in square centimetres. If your answer has the wrong units, you used the wrong formula.
The pizza question
A 16-inch pizza costs a bit less than two 8-inch ones. Which is more food?
The 16-inch, by a lot. Its radius is twice, so its area is FOUR times — 201 square inches against 50. Two 8-inch pizzas give you 100. The single large one gives you double that.
The general rule: area scales with the square of the radius. A 20% bigger diameter is 44% more pizza. An 18-inch is 27% larger than a 16-inch, not 12.5%.
This is the most-used circle fact in ordinary life, and almost nobody applies it. It works identically for round tables, cake tins and pipe cross-sections — a pipe of twice the diameter carries four times the flow at the same velocity.
A semicircle is not half a circumference
Half the AREA is half the area. Half the PERIMETER is not half the perimeter.
Cut a circle in half and the curved part is indeed half the circumference, πr. But the cut edge — the diameter, 2r — is now part of the boundary. The semicircle’s perimeter is πr + 2r, which is about 64% of the full circumference, not 50%.
It matters wherever the boundary is what you are buying: edging for a semicircular flowerbed, trim round an arched window, fencing for a D-shaped enclosure. Halving the circumference underestimates by the whole diameter.
The page reports both, separately labelled, for exactly this reason.
Circles and squares
A circle fills exactly π/4 of the square around it — 78.54% — whatever its size. The four corners are the other 21.46%.
That constant ratio is why cutting circles from square sheet always wastes about a fifth of the material, and why hexagonal packing is used where the waste matters.
Going the other way: the largest square that FITS INSIDE a circle has its diagonal equal to the diameter, so its side is d/√2. It covers about 63.7% of the circle.
Both figures are size-independent, which is what makes them worth knowing rather than recomputing.
What π actually is
π is the ratio of any circle’s circumference to its diameter. Every circle, of every size, gives the same number — that constancy is the fact, and it is not obvious.
It is irrational: no fraction equals it exactly, and its decimal expansion never repeats. 22/7 is accurate to two decimal places, 355/113 to six — good enough for almost anything and still not exact.
It is also transcendental, which is the stronger statement that it is not the root of any polynomial with whole-number coefficients. That result, proved by Lindemann in 1882, is what settled the ancient problem of squaring the circle: it cannot be done with compass and straightedge.
For practical work, 3.14159 is plenty. NASA uses about fifteen decimal places for interplanetary navigation, and forty would locate a point on the edge of the observable universe to within the width of a hydrogen atom.
Where the asymmetry bites
Circumference is linear in the radius; area is quadratic. Most circle errors are one of those being used where the other belongs.
Buying by area. Pizza, cake, table tops, floor space. A modest increase in diameter is a large increase in area.
Buying by circumference. Edging, trim, fencing, belt length. These scale linearly, so doubling the size doubles the cost rather than quadrupling it.
Flow through a pipe. Capacity goes with the cross-sectional AREA, so a pipe of twice the diameter carries four times as much. Sizing one by diameter alone underestimates badly.
Watering and spraying. Coverage is an area, so doubling the reach of a sprinkler quadruples the ground it waters — and quarters the depth it delivers.
Sources and methodology
The constant and the conventions are standardised; these are the references.
Method. Whichever measurement is supplied is converted to a canonical radius, and the other three are derived from that single value — so the four can never disagree with each other, which is the property the suite checks by round-tripping every entry point. The semicircle perimeter is computed as the arc plus the diameter rather than as half the circumference, and the suite asserts they are never equal. That engine is verified on every change against 76 hand-written assertions, including that every entry point recovers the same radius across nine thousand round trips, and that the circle covers exactly π/4 of its circumscribed square at every size. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Central AngleLeave one of radius, arc and angle blank and the page solves for it — with the sector drawn, plus chord, sector area and segment area from the same angle.
Pythagorean Theorema² + b² = c² solved for whichever side you leave blank, with the triangle drawn to scale and integer triples reduced to their primitive.
TriangleSolve any triangle from three measurements — and when two sides and a non-included angle describe TWO triangles, this one shows both instead of picking one.
Heron’s FormulaArea from three sides in the rearrangement that keeps its digits — with the textbook form shown beside it so you can see what the usual order costs.
Square RootThe exact square root first — 72 gives 6 root 2 — then the decimal to as many as sixty places, computed on whole numbers rather than a double.
Similar TrianglesScale factor and the missing sides — plus the area factor, which is its square. Two pairs with different ratios are reported as not similar rather than averaged.
For part of a circle rather than all of it — an arc, a sector or a segment — the Central Angle Calculator solves those from the radius and the angle.
Educational use disclaimer
This is an educational tool. π is irrational, so the decimal results are rounded; what is exact is the consistency, since all four measurements are derived from one radius.
Published the circle page deriving all four measurements from one canonical radius, so they can never disagree with each other whichever one you start from.
Built around the asymmetry that causes most circle errors: circumference is linear in the radius and area is quadratic, so a 16-inch pizza is four times an 8-inch and not twice.
Reports the semicircle perimeter as the arc PLUS the diameter rather than half the circumference, which under-orders edging by the whole diameter.
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