Square In A Circle Calculator

Background.

What is the biggest square that fits inside a circle? Rotate any inscribed square and its four corners ride the circle while its diagonals pass through the centre — which is the entire solution: the square's diagonal is a diameter. From diagonal = 2r and the square's own diagonal-to-side ratio of √2, the side is r√2 and the area is 2r². This page computes both from the radius.

The ratio behind the answer is worth pausing on. A square of side s has diagonal s√2 — Pythagoras on two equal legs — and that √2 ≈ 1.414 is the constant that governs every fit-a-square problem. Here it works in reverse: the circle fixes the diagonal, so the side is the diameter shrunk by √2, and the square claims area 2r² out of the circle's πr² — a fixed 2/π ≈ 63.7% of the disc, independent of size. The four leftover circular segments split the remaining 36.3%.

The question is practical more often than it looks. Machinists milling the largest square boss from round bar stock, woodworkers sawing a square post from a log, engineers windowing a square sensor inside a round lens' image circle, and fabricators cutting square blanks from circular offcuts all face exactly this geometry — usually meeting it as ‘what square section does a 100 mm round yield?’ The answer, 70.7 mm on a side, is this page's arithmetic.

The model is the ideal centred, fully inscribed square — corners exactly on the circle. Real cutting adds kerf, clearance, and tolerance allowances that only your process can supply, as the scope note beside the result records.

What is square in a circle calculator?

The inscribed square of a circle is the largest square that fits inside it: all four corners lie on the circle, and its diagonals are diameters. The geometry gives closed forms from the radius alone — side = r√2, area = 2r² — because the diagonal (2r) and the side of a square are locked in the ratio √2. The inscribed square occupies exactly 2/π ≈ 63.66% of the circle's area at every scale. It is the reverse of the circumscribed-square problem (square outside, circle inside), whose square has side 2r and area 4r² — exactly twice the inscribed square's.

How to use this calculator.

1. 01Enter the circle's radius — halve a measured diameter first, since stock and holes are usually specified by diameter.
2. 02Read the inscribed square's side (r√2) and area (2r²), in your unit and unit-squared respectively.
3. 03For machining or sawing, subtract your process allowances from the side afterwards — kerf, cleanup passes, and tolerance are not in the ideal geometry.
4. 04Going the other way — what circle does a given square need? — invert to r = s/√2: a 50 mm square needs at least a 70.7 mm-diameter circle.
5. 05Sanity-check any answer with the area ratio: the square should come out to just under two-thirds (63.7%) of the circle's area, every time.

The formula.

Two facts assemble the result. First, the largest inscribed square is centred, so each diagonal runs corner-to-corner through the centre — a chord through the centre is a diameter, hence diagonal d = 2r. (Thales' theorem gives the same from the other side: the right angle at each corner subtends a diameter.) Second, a square's diagonal and side obey d = s√2, from Pythagoras on the right triangle formed by two sides and the diagonal. Combining: s = 2r/√2 = r√2, and squaring gives area = 2r² — with the tidy check that the four corner segments left over must total πr² − 2r². The fixed fraction 2/π falls out immediately: inscribed square over circle = 2r²/πr², scale-free. The same two facts answer the family of related questions — circumscribed square (side 2r), inscribed circle of a square (radius s/2), largest square in a semicircle (a different, r√(4/5) problem worth not confusing with this one). The engine multiplies r by √2 and squares in Decimal arithmetic, rounding once to twelve significant digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06. independence: primary; access: open.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06. independence: secondary-check; access: open.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06. independence: primary; access: open.