1. What is Space Diagonal of Cube given Insphere Radius?

The space diagonal of a cube is the longest distance between any two opposite vertices of the cube. When given the insphere radius (the radius of the largest sphere that fits inside the cube), we can calculate this diagonal using a specific mathematical relationship.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( \sqrt{3} \) — Square root of 3 (approximately 1.732)
• Insphere Radius of Cube — Radius of the sphere inscribed in the cube

Explanation: This formula derives from the geometric relationship between the insphere radius and the cube's dimensions, where the insphere radius equals half the cube's side length.

3. Importance of Space Diagonal Calculation

Details: Calculating the space diagonal is important in geometry, architecture, and 3D modeling for determining maximum distances within cubic structures and ensuring proper spatial relationships in design.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the space diagonal of the cube.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between insphere radius and cube side length?
A: The insphere radius equals half the side length of the cube (\( r_i = a/2 \)).

Q2: How is the space diagonal related to the side length?
A: The space diagonal equals the side length multiplied by \( \sqrt{3} \) (\( d = a\sqrt{3} \)).

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to cubes where all faces are squares and all angles are right angles.

Q4: What are practical applications of space diagonal calculation?
A: Used in packaging design, construction, 3D graphics, and determining maximum material lengths that can fit inside cubic containers.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, though practical measurements may have precision limitations based on measurement tools.