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When a square is inscribed in a circle (meaning all four of the square's corners touch the circle's edge), the square's diagonal is exactly equal to the circle's diameter: 2r = s√2. This calculator converts between the square's side length and the circle's radius using that relationship, and is a common building block in geometry problems, in mechanical design (fitting a square shaft or bracket within a circular housing), and in packing/layout problems where a square component must fit exactly within a circular boundary.
How it works
Enter either the square's side length or the circle's radius. The relationship comes directly from the Pythagorean theorem: the square's diagonal d satisfies d² = s² + s² = 2s², so d = s√2. Since that diagonal is also the circle's diameter (2r), we get 2r = s√2, which rearranges to r = s√2/2 (equivalently s/√2), or solved the other way, s = r√2 = 2r/√2. The calculator applies whichever direction you need and also reports the diagonal (equal to 2r) and the area of both shapes.
- Enter given value.
- Enter value.
- Click Calculate to see your results.
Examples
Side = 10
Radius = 10√2/2 = 5√2 ≈ 7.071, diagonal = 10√2 ≈ 14.142.
Radius = 5
Side = 5√2 ≈ 7.071, diagonal = 2 × 5 = 10.
Side = 6
Radius = 6√2/2 = 3√2 ≈ 4.243, diagonal = 6√2 ≈ 8.485.
Who should use it
- Finding the largest square that fits inside a circular opening.
- Geometry coursework on inscribed shapes.
- Mechanical design problems fitting a square component within a circular boundary.
Industry applications
- Geometry education
- Mechanical and industrial design
Advantages
- Converts in both directions between side length and radius.
- Also reports diagonal and both shapes' areas in one step.
Limitations
- Only covers the specific inscribed-square-in-circle configuration, not a circle inscribed in a square or other polygon-circle combinations.
Common mistakes to avoid
- Confusing an inscribed square (corners touch the circle) with a circle inscribed in a square (circle touches the square's sides) — these describe different shapes and use different formulas.
- Forgetting to account for the √2 factor and assuming the side length simply equals the radius or diameter.
- Mixing up the diagonal (s√2) with the side length (s) when reading off results.
Best practices
- Double check which configuration your problem actually describes — square-in-circle or circle-in-square — before choosing a formula.
- Use the diagonal = diameter shortcut as a quick way to verify your answer by hand.
- Remember the area ratio (2/π ≈ 63.7%) as a useful sanity check on computed areas.
Tips
- Need the reverse configuration (a circle inscribed inside a square) instead? That uses diameter = side length directly, a simpler relationship without the √2 factor.