What this calculator does
This circle square calculator answers two related questions that people search for interchangeably: how big a square can you cut from a circle of a given size, and how big a circle do you need to fully enclose a square you already have. Both come from the same geometric fact, so one calculator handles either direction.
The relationship hinges on the diagonal. A square inscribed in a circle touches the circle at all four corners, which means the square's diagonal is exactly the circle's diameter. Once you fix that, the side length of the square and the radius of the circle are locked to each other by a single constant, the square root of two.
The formula
For a square inscribed in a circle, the diagonal of the square equals the diameter of the circle (2 × radius). Since a square's diagonal is its side times the square root of two, the side length works out to radius × √2. Run the same relationship backwards, starting from a square's side, and the smallest circle that passes through all four corners has a radius of side × √2 ÷ 2.
| Term | Meaning |
|---|---|
| Radius | Distance from the centre of the circle to its edge. |
| Side | Length of one edge of the square. |
| Diagonal | The line from one corner of the square to the opposite corner, equal to the circle's diameter when the square is inscribed. |
| √2 | The square root of two, roughly 1.41421, the fixed constant linking a square's side to its diagonal. |
The inputs explained
| Field | What to enter |
|---|---|
| Find | Choose whether you are starting from a circle (find the biggest square that fits inside it) or a square (find the smallest circle that fits around it). |
| Circle radius (circle-to-square mode) (cm) | The circle's radius, used only in circle-to-square mode. |
| Square side (square-to-circle mode) (cm) | The square's side length, used only in square-to-circle mode. |
When to use it
Cutting a square panel from round stock
Someone with a circular sheet of material, such as glass, timber or metal, wants to know the largest square panel that can be cut from it without running off the edge.
Framing a square object inside a round opening
A square photo, tile or hatch needs to fit through or be mounted inside a circular opening, and the smallest suitable circle (or the largest square that clears a given circle) needs checking before cutting anything.
Checking a design constraint quickly
A drawing shows a circle and a square meant to relate to each other, and a quick sanity check confirms whether the dimensions given actually satisfy that inscribed relationship.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the largest square that fits inside circles of common radii?
The square-to-circle relationship stays proportional, so scaling the radius scales the fitted square by the same factor.
| Circle radius | Square side length | Square area |
|---|---|---|
| 1 units | 1.41 units | 2.00 sq units |
| 2 units | 2.83 units | 8.00 sq units |
| 5 units | 7.07 units | 50.00 sq units |
| 10 units | 14.14 units | 200.00 sq units |
| 20 units | 28.28 units | 800.00 sq units |
| 50 units | 70.71 units | 5,000.00 sq units |
What size circle do you need to fit around a square?
Starting from the square's side instead, the smallest enclosing circle scales the same way in reverse.
| Square side | Circle radius | Circle diameter |
|---|---|---|
| 1 units | 0.71 units | 1.41 units |
| 2 units | 1.41 units | 2.83 units |
| 5 units | 3.54 units | 7.07 units |
| 10 units | 7.07 units | 14.14 units |
| 20 units | 14.14 units | 28.28 units |
| 50 units | 35.36 units | 70.71 units |
Questions
What is the formula for a square circle relationship like this?
For a square inscribed in a circle, side = radius × √2. For the reverse, a circle drawn around a square, radius = side × √2 ÷ 2. Both come from the square's diagonal being equal to the circle's diameter.
Is the largest square in a circle always centred?
Yes. The largest square that fits inside a circle without crossing the edge is inscribed with all four corners touching the circle, and this only works if the square is centred on the circle's centre.
Does this work the same for a circle inscribed inside a square instead?
That is a different relationship. A circle inscribed inside a square, touching the midpoint of each side, has a diameter equal to the square's side, not its diagonal. This calculator covers the corner-touching case in both directions, since that is the one usually meant by circle square problems.
Can I use this for any unit of length?
Yes. The radius and side length are entered as plain numbers, so use millimetres, inches, metres or any consistent unit; the result comes back in the same unit.
For the area or perimeter of a plain circle or square on their own, without the inscribed relationship, see Area of common shapes or Perimeter of common shapes.