1. What is the Largest Square in a Circle?

The largest square that fits inside a circle is the square whose diagonal is equal to the diameter of the circle. This relationship is fundamental in geometry and has applications in various fields including engineering and design.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

• \( r \) — Radius of the circle (same units as result)
• \( \sqrt{2} \) — Approximately 1.4142 (constant)

Explanation: The diagonal of the square equals the diameter of the circle (2r). Using the Pythagorean theorem for squares (diagonal = side × √2), we derive the side length.

3. Practical Applications

Details: This calculation is used in construction (determining largest square beam that fits in circular pipe), manufacturing (cutting square pieces from circular stock), and graphic design (fitting square elements in circular spaces).

4. Using the Calculator

Tips: Simply enter the radius of your circle in any length units. The result will be in the same units. The radius must be a positive number.

5. Frequently Asked Questions (FAQ)

Q1: Why does the formula use √2?
A: The √2 comes from the Pythagorean theorem applied to a square, where diagonal = side × √2. Since the diagonal equals the circle's diameter, we solve for the side length.

Q2: Can I use diameter instead of radius?
A: Yes, but you would need to divide your diameter by 2 first, or use the formula: side = diameter / √2.

Q3: What's the area of the largest square?
A: The area would be side² = (r√2)² = 2r², which is about 63.7% of the circle's area.

Q4: Does this work for 3D (cube in sphere)?
A: No, this is a 2D calculation. For a cube in a sphere, the space diagonal equals the sphere's diameter.

Q5: What's the relationship to inscribed squares?
A: This is specifically about the largest possible square that can be inscribed in a circle, which is unique up to rotation.