How it works
Square in circle: s = d ÷ √2 = r√2, square area = 2r². Circle around square: d = s√2. Circle in square: d = s.
- Choose the arrangement you need.
- Enter the circle radius or diameter, or the square side.
- Choose a length unit and compare the areas and fill percentage.
Worked example: A circle of radius 5 cm fits a square with side 5√2 cm, about 7.0711 cm. The square area is 50 cm² and the circle area is 78.5398 cm². The square fills 63.66% of the circle, leaving 28.5398 cm².
Example square and circle fits
| Arrangement | Square side | Square area | Circle area | Square fill of circle | Measurement |
|---|---|---|---|---|---|
| Square inside circle | 7.0711 cm (5√2 cm) | 50 cm² | 78.5398 cm² | 63.66% | r 5 cm |
| Square inside circle | 7.0711 cm (5√2 cm) | 50 cm² | 78.5398 cm² | 63.66% | d 10 cm |
| Square inside circle | 21.2132 cm (15√2 cm) | 450 cm² | 706.8583 cm² | 63.66% | d 30 cm |
| Circle around square | 10 cm | 100 cm² | 157.0796 cm² | 63.66% | Side 10 cm |
| Circle inside square | 10 cm | 100 cm² | 78.5398 cm² | 78.54% | Side 10 cm |
Swipe the table sideways to see the rest.
Things to consider
- For square timber from a log, use the usable log diameter.
- Allow for saw cuts, bark and defects.
- A square peg fits a round hole when its diagonal fits the diameter.
- Leave clearance for tiles, tables and pipe openings.
Frequently asked questions
What is the formula for a square inscribed in a circle?
The largest square has side s = d ÷ √2 = r√2. Its diagonal equals the circle diameter and its area is 2r². For circle area alone, use the Area of a Circle Calculator.
Why does the square diagonal equal the circle diameter?
Opposite corners of an inscribed square lie on opposite points of the circle. The diagonal passes through the centre, so it is a diameter. Pythagoras gives diagonal = √(s² + s²) = s√2.
What fraction of the circle does the square fill?
The square area is 2r² and the circle area is πr². Their ratio is 2/π, about 63.66%. A circle inside a square instead fills π/4, about 78.54% of the square.
What do inscribed and circumscribed mean?
An inscribed shape fits inside another shape and touches its boundary. A circle circumscribed around a square passes through its four corners. A circle inscribed in a square touches the middle of each side.
How much square timber can I cut from a round log?
For an ideal circular log, divide the usable diameter by √2 to get the largest square side. A 30 cm diameter gives about 21.2132 cm. Actual timber needs allowances for bark, defects, saw cuts and variation along the log.
How do I fit a square peg, tile or table in a round area?
The square's diagonal must be no greater than the round opening's diameter. This applies to an ideal square peg in a round hole or a square tile or table inside a circular area. Leave practical clearance. The Rectangle Calculator: Area and Perimeter can help with a non-square footprint, and the Circumference Calculator measures the circular boundary.
What size round pipe fits through a square opening?
A centred round pipe can have an outside diameter up to the square side in the ideal fit. Use Circle inside square and enter the opening's side. Leave clearance for installation, tolerances and fittings.
Results are estimates for general information only, not financial, medical or professional advice.