Circumsphere Radius of Stellated Octahedron given Total Surface Area Calculator

Formula Used:

\[ r_c = \frac{\sqrt{6}}{4} \times \sqrt{\frac{2 \times TSA}{3 \times \sqrt{3}}} \]

Circumsphere Radius of Stellated Octahedron:

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1. What is Circumsphere Radius of Stellated Octahedron?

The Circumsphere Radius of a Stellated Octahedron is the radius of the sphere that contains the Stellated Octahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property in three-dimensional geometry.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{6}}{4} \times \sqrt{\frac{2 \times TSA}{3 \times \sqrt{3}}} \]

Where:

\( r_c \) — Circumsphere Radius of Stellated Octahedron (m)
\( TSA \) — Total Surface Area of Stellated Octahedron (m²)
\( \sqrt{6} \) — Square root of 6 (approximately 2.44949)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the circumsphere radius based on the total surface area of the stellated octahedron, using mathematical constants and geometric relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions of the stellated octahedron, its relationship with enclosing spheres, and for various applications in geometry, architecture, and 3D modeling.

4. Using the Calculator

Tips: Enter the total surface area of the stellated octahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Stellated Octahedron?
A: A stellated octahedron is a polyhedron created by extending the faces of a regular octahedron until they meet again, forming a star-shaped three-dimensional figure.

Q2: What units should I use for the input?
A: The calculator expects the total surface area in square meters (m²), and returns the circumsphere radius in meters (m).

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the stellated octahedron only. Other polyhedra have different formulas for calculating circumsphere radius.

Q4: What is the significance of the circumsphere?
A: The circumsphere is the smallest sphere that can completely contain the polyhedron, with all vertices touching the sphere's surface.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the given formula, assuming accurate input values and proper implementation of the mathematical operations.