Circumsphere Radius of Disheptahedron given Surface to Volume Ratio Calculator

Formula Used:

\[ r_c = \frac{6 \times (3 + \sqrt{3})}{5 \times \sqrt{2} \times \frac{R_A}{V}} \]

Circumsphere Radius (r_c):

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1. What is Circumsphere Radius of Disheptahedron?

The Circumsphere Radius of a Disheptahedron is the radius of the sphere that contains the Disheptahedron in such a way that all the vertices of the Disheptahedron are touching the sphere. It represents the smallest sphere that can completely enclose the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{6 \times (3 + \sqrt{3})}{5 \times \sqrt{2} \times \frac{R_A}{V}} \]

Where:

\( r_c \) — Circumsphere Radius (m)
\( \frac{R_A}{V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{2} \) — Square root of 2 (approximately 1.414)

Explanation: This formula calculates the circumsphere radius based on the surface to volume ratio of the disheptahedron, using mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is important in geometry and materials science for understanding the spatial dimensions and packing efficiency of polyhedral structures. It helps in determining the minimum bounding sphere required to contain the polyhedron.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Disheptahedron?
A: A Disheptahedron is a polyhedron with 14 faces, typically consisting of a combination of triangular and square faces in a specific geometric arrangement.

Q2: How is Surface to Volume Ratio defined?
A: Surface to Volume Ratio is the ratio of the total surface area of a polyhedron to its volume, measured in m⁻¹ (inverse meters).

Q3: What are typical values for Circumsphere Radius?
A: The circumsphere radius varies depending on the size and proportions of the disheptahedron, but it's always larger than or equal to the inradius.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the disheptahedron geometry and may not apply to other polyhedral shapes.

Q5: What are the practical applications of this calculation?
A: This calculation is used in crystallography, materials science, and geometric modeling to understand the spatial properties and packing characteristics of disheptahedral structures.