Circumsphere Radius of Truncated Rhombohedron given Area of Pentagon Calculator

Formula Used:

\[ r_c = \frac{\sqrt{14 - 2\sqrt{5}}}{4} \times \sqrt{\frac{4 \times A_{\text{Pentagon}}}{\sqrt{5 + 2\sqrt{5}}}} \]

Circumsphere Radius of Truncated Rhombohedron:

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1. What is Circumsphere Radius of Truncated Rhombohedron?

The Circumsphere Radius of a Truncated Rhombohedron is the radius of the sphere that contains the polyhedron in such a way that all its vertices lie on the surface of the sphere. It represents the smallest sphere that can completely enclose the truncated rhombohedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{14 - 2\sqrt{5}}}{4} \times \sqrt{\frac{4 \times A_{\text{Pentagon}}}{\sqrt{5 + 2\sqrt{5}}}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( A_{\text{Pentagon}} \) — Area of Pentagon of Truncated Rhombohedron (m²)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula calculates the circumsphere radius based on the area of the pentagonal faces of the truncated rhombohedron, utilizing geometric relationships and mathematical constants.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry and 3D modeling for determining the bounding sphere of complex polyhedra, which is useful in collision detection, spatial analysis, and computer graphics applications.

4. Using the Calculator

Tips: Enter the area of the pentagon in square meters. The value must be positive and greater than zero. The calculator will compute the circumsphere radius based on the geometric properties of the truncated rhombohedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Rhombohedron?
A: A truncated rhombohedron is a polyhedron obtained by cutting the corners of a rhombohedron, resulting in a shape with pentagonal and hexagonal faces.

Q2: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of a perfect truncated rhombohedron, assuming precise input values.

Q3: What units should I use for the area?
A: The area should be in square meters (m²) for consistent results. The calculator will return the radius in meters.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated rhombohedra. Other polyhedra have different geometric relationships and require different formulas.

Q5: What if I have the edge length instead of pentagon area?
A: You would need to first calculate the pentagon area from the edge length using the appropriate formula before using this calculator.