1. What is the Largest Square Inside a Circle?

The largest square that can fit inside a circle is called an inscribed square, where all four vertices lie exactly on the circumference of the circle. This geometric relationship is important in various fields including architecture and engineering.

2. How Does the Calculator Work?

The calculator uses the simple formula:

Where:

• \( r \) — Radius of the circle
• \( \sqrt{2} \) — Approximately 1.4142

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

3. Geometric Relationship

Details: The area of the inscribed square is exactly half the area of the circumscribed circle. This relationship holds true regardless of the circle's size.

4. Using the Calculator

Tips: Simply enter the radius of your circle in any length units. The calculator will output the side length of the largest possible square that fits inside in the same units.

5. Frequently Asked Questions (FAQ)

Q1: Why is √2 involved in this calculation?
A: The √2 comes from the Pythagorean theorem applied to the square's diagonal, which must equal the circle's diameter.

Q2: What's the area of the inscribed square?
A: Square area = side² = (r√2)² = 2r², which is exactly half the circle's area (πr²).

Q3: Can this be extended to 3D (cube in sphere)?
A: Yes! For a cube inscribed in a sphere, the cube's space diagonal equals the sphere's diameter, giving side = (2r)/√3.

Q4: What about other regular polygons?
A: Each regular polygon has its own relationship between side length and circumradius. For example, an equilateral triangle has side = r√3.

Q5: How precise is this calculation?
A: Mathematically exact, though practical measurements may have precision limitations based on your instruments.