Square in a Circle Calculator

A square is inscribed in a circle when all four corners of the square touch the circle. This arrangement creates a direct relationship between the circle's radius and every measurement of the square. The square in a circle calculator uses that relationship to find the maximum square side, area, perimeter, and diagonal from one radius. It is useful in geometry, machining, packaging, graphic design, architecture, and material cutting whenever a square must fit inside a circular boundary. The key observation is that the diagonal of the inscribed square passes through the center of the circle and connects two opposite points on the circumference. It is therefore exactly equal to the circle's diameter, which is twice the radius. Splitting the square along that diagonal produces a right isosceles triangle. By the Pythagorean theorem, the diagonal equals the side multiplied by the square root of 2. Rearranging gives the side as the diameter divided by the square root of 2, or equivalently the radius multiplied by the square root of 2. Once the side is known, the remaining properties follow from ordinary square formulas. Area is side squared, which simplifies to two times the radius squared. Perimeter is four times the side, or four times the radius times the square root of 2. The diagonal remains two times the radius. For a radius of 5 units, the inscribed square has a side of about 7.0711 units, an area of 50 square units, a perimeter of about 28.2843 units, and a diagonal of 10 units. The computed square is the largest square that can be placed inside the given circle when centered. A smaller square can fit without all corners touching, but it is not inscribed in the strict geometric sense. Rotating a centered square does not change the required circle because every corner stays the same distance from the center. The formula also assumes an ideal circle and square; physical fabrication may require clearance or a tolerance reduction. Enter measurements in any consistent unit. If the radius is in millimeters, the side, perimeter, and diagonal are in millimeters while the area is in square millimeters. If you know only the diameter, divide it by two before entering the radius. The calculator preserves numerical precision and rounds only the displayed values, making it suitable both for quick estimates and for checking hand calculations.