Circumscribed Cylinder Radius of Cube given Space Diagonal Calculator

Formula Used:

\[ r_c = \frac{d_{space}}{\sqrt{6}} \]

Circumscribed Cylinder Radius of Cube:

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1. What is Circumscribed Cylinder Radius of Cube?

The Circumscribed Cylinder Radius of Cube is the radius of the cylinder that contains the Cube in such a way that all the vertices of the Cube are touching the cylinder. This represents the smallest cylinder that can completely enclose a given cube.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{d_{space}}{\sqrt{6}} \]

Where:

\( r_c \) — Circumscribed Cylinder Radius of Cube (m)
\( d_{space} \) — Space Diagonal of Cube (m)
\( \sqrt{6} \) — Mathematical constant (approximately 2.44949)

Explanation: The formula derives from the geometric relationship between the space diagonal of a cube and the radius of the circumscribed cylinder that touches all eight vertices of the cube.

3. Importance of Circumscribed Cylinder Radius Calculation

Details: Calculating the circumscribed cylinder radius is important in various engineering and design applications where cubes need to be fitted into cylindrical containers or enclosures, particularly in manufacturing, packaging, and architectural design.

4. Using the Calculator

Tips: Enter the space diagonal of the cube in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between cube side length and circumscribed cylinder radius?
A: Since space diagonal \( d = s\sqrt{3} \), the circumscribed cylinder radius can also be expressed as \( r_c = \frac{s\sqrt{3}}{\sqrt{6}} = s\sqrt{\frac{1}{2}} \), where s is the side length of the cube.

Q2: How does this differ from inscribed cylinder radius?
A: The circumscribed cylinder touches all vertices of the cube, while the inscribed cylinder touches the midpoints of the cube's faces. The inscribed cylinder has a smaller radius.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to cubes. Other polyhedra have different relationships between their space diagonals and circumscribed cylinder radii.

Q4: What are practical applications of this calculation?
A: This calculation is useful in mechanical engineering for designing cylindrical housings for cubic components, in packaging design for cylindrical containers holding cubic items, and in architectural design.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, provided accurate input values are given. The result's practical accuracy depends on the precision of the space diagonal measurement.