Circular Square Calculator

Name: Circular Square - Geometry Calculator
Author: Jürgen Kummer

Calculations at a circular square. Here, circular square is used as name for a shape which occurs, if from a circle an overlapping cross with equally thick bars is removed centrally and the four remaining parts are jointed. The four vertices can be connected to a square, the edges are arcs of the original circle.
Enter the circle radius and the bar thickness. Choose the number of decimal places, then click Calculate. The bar thickness must be lower than √2 times the radius.

Construction of a circular square. The cross is removed from the circle, the four parts will be jointed.

Formulas:

h = r - \frac{\sqrt{4 r^{2} - b^{2}}}{2}

a = \sqrt{2} \cdot (r - \frac{b}{2} - h)

l = r \cdot [\frac{π}{2} - 2 \cdot \arccos (1 - \frac{h}{r})]

d_{1} = 2 r - b - 2 h

d_{2} = 2 r - \sqrt{2} \cdot b

p = 4 l

A = π r^{2} - 4 b r + b^{2} + 4 b h - 2 [r^{2} \cdot 2 \cdot \arccos (1 - \frac{h}{r}) - b (r - h)]

pi:

π = 3.141592653589793...

Radius, width, height, length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The four parts that make up this shape each consist of a circular segment with radius r and chord a, as well as an isosceles right triangle with hypotenuse a and legs d₁ / 2. The thicker the bars of the cross that is removed, the smaller the circular square becomes and the more pointed its corners become.

The circular square has the same symmetry properties as the square. It is point-symmetric about its center and rotationally symmetric about a rotation of 90 degrees and multiples thereof around this center. It is also axially symmetric about a total of four axes. Two axes of symmetry pass through opposite vertices, and two axes of symmetry pass through the midpoints of opposite sides, which here are circular arcs.

The circular square is a round, convex solid with vertices and no straight sides. This is something it has in common with the pointed oval and the Reuleaux triangle.

↑ up