1. What is a Square Inscribed in Circle?

A square inscribed in a circle is a square drawn inside a circle such that all four vertices of the square lie on the circumference of the circle. The circle is called the circumcircle of the square.

2. How Does the Calculator Work?

The calculator uses these key formulas:

Where:

• \( r \) — Radius of the circumscribed circle
• Diagonal — Length between opposite vertices of the square
• Side — Length of one edge of the square

Explanation: The diagonal of the inscribed square equals the diameter of the circle (2r). The side length relates to the radius through the Pythagorean theorem.

3. Importance of These Calculations

Details: These calculations are fundamental in geometry, useful in architecture, engineering, and design where precise measurements of inscribed shapes are needed.

4. Using the Calculator

Tips: Simply enter the radius of the circle to calculate both the diagonal and side length of the inscribed square. The radius must be a positive number.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between circle diameter and square diagonal?
A: They are equal - the diagonal of the inscribed square is exactly the same length as the diameter of the circumscribed circle.

Q2: How is the side length derived from the radius?
A: Using the Pythagorean theorem: side² + side² = (2r)² → 2×side² = 4r² → side = r√2.

Q3: Can this calculator work in reverse?
A: Yes, if you know the square's side length, the circle's radius is side/√2. If you know the diagonal, the radius is diagonal/2.

Q4: What are practical applications of this?
A: Useful in construction (circular windows with square frames), manufacturing (turning circles into squares), and graphic design.

Q5: Does this work for rectangles inscribed in circles?
A: No, this specific calculator only works for squares. Rectangles have different geometric relationships.