Calculate properties of a square inscribed in a circle. Find diagonal, side length, areas of both shapes, area ratio (2/π), and wasted space from radius, side, or diagonal.
A square inscribed in a circle is one of geometry's most elegant configurations. The square's four corners touch the circle, and its diagonal equals the circle's diameter. This relationship appears everywhere—from engineering tolerances for round-to-square transitions, to packaging design, CNC machining, and math competitions.
This Square in a Circle Calculator lets you start from any known measurement—circle radius, square side length, or square diagonal—and instantly derives every other property. You'll see the square's side and diagonal, both areas, the constant area ratio of 2/π ≈ 0.6366, the wasted area (the crescent-shaped corners), and perimeters for both shapes.
The area ratio is always the same regardless of size: a square inscribed in a circle always covers about 63.66 % of the circle's area. The remaining 36.34 % is the four corner regions between the square and the circle. The calculator's visual bars give you an intuitive sense of this relationship, while the reference table shows exact values for common radii from 1 to 50 units. Whether you're solving a geometry problem, sizing a square beam inside a round pipe, or designing a coaster, this tool makes the math instant and transparent.
This calculator is useful when you need more than a single side-length conversion. Starting from a radius, side, or diagonal, it shows the full geometry of the inscribed square in one place: side, diagonal, both areas, the fixed area ratio of 2/π, and how much of the circle remains outside the square. That makes it practical for classroom geometry, design layouts, machining clearances, and checking whether a square part will fit inside a round opening.
Side = r × √2. Diagonal = 2r. Square area = 2r². Circle area = πr². Area ratio = 2/π ≈ 0.6366. Wasted area = r²(π − 2).
Result: For side 20, the circle radius is about 14.1421 and the square covers 63.66% of the circle.
Using Square Side mode with a side length of 20 gives radius = 20 / √2 ≈ 14.1421 and diagonal = 28.2843. The square area is 400, the circle area is about 628.3185, the area ratio stays 2/π ≈ 0.6366, and the wasted corner area is about 228.3185.
The defining characteristic of a square inscribed in a circle is an area ratio of 2/π ≈ 0.6366, which holds for any circle size. The inscribed square has side = r√2 and area = 2r². The enclosing circle has area = πr². Their ratio is 2r² / (πr²) = 2/π—a universal geometric constant. The remaining four curved corner regions collectively cover (1 − 2/π) ≈ 36.34% of the circle's area. No matter how large or small the circle, the inscribed square always fills exactly 63.66% of it.
The four corners of an inscribed square lie exactly on the circle. Two opposite corners are diametrically opposite—they are endpoints of a diameter. Therefore, the diagonal of the inscribed square equals the diameter of the circle. This follows from Thales' theorem: any angle inscribed in a semicircle that subtends the diameter is exactly 90°. Since a square's corner angle is always 90°, opposite corners of the inscribed square must define a diameter. Conversely, given the diagonal, the circumscribed circle's diameter is known immediately.
- **Square stock inside round tubing:** A round steel tube with 2-inch inner diameter can hold a square rod no larger than side = 2/√2 ≈ 1.414 inches—the largest square cross-section that fits without rotation. - **CNC machining:** When milling a square pocket inside a circular bore, the inscribed-square formula sets the maximum workpiece width for any given bore diameter. - **Packaging and coasters:** A square lid fitted inside a circular container has diagonal = container inner diameter; the formula gives the exact side length to cut. - **Logo and graphic design:** Inscribing a square in a circle creates a balanced composition; the ratio 2/π ≈ 0.637 is close to the golden ratio, contributing to its aesthetic appeal.
The circumscribed square (the circle sits inside the square) has side = 2r and area = 4r². The inscribed-to-circumscribed area ratio is exactly 1/2. The circle's area πr² sits between them: inscribed square area : circle area : circumscribed square area = 2 : π : 4, an elegant triple ratio that expresses how the circle is geometrically sandwiched between the two squares.
A square inscribed in a circle has all four vertices touching the circle. The square's diagonal equals the circle's diameter.
The ratio is always 2/π ≈ 0.6366. The inscribed square covers about 63.66 % of the circle's area.
Multiply the radius by √2. For example, radius 10 gives side = 10 × √2 ≈ 14.142.
Because opposite vertices of the inscribed square are endpoints of a diameter—the diagonal passes through the center of the circle. Use this as a practical reminder before finalizing the result.
About 36.34 % of the circle's area lies in the four corner regions outside the square. In formula form: wasted = r²(π − 2).
Yes. Select "Square Side Length" mode and the calculator derives the radius as r = side / √2.