Circumsphere Radius of Rotunda given Total Surface Area Calculator

Formula Used:

\[ r_c = \frac{1}{2} \times (1 + \sqrt{5}) \times \sqrt{\frac{TSA}{\frac{1}{2} \times ((5 \times \sqrt{3}) + \sqrt{10 \times (65 + (29 \times \sqrt{5}))})}} \]

Circumsphere Radius of Rotunda (r_c):

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1. What is the Circumsphere Radius of Rotunda?

The Circumsphere Radius of Rotunda is the radius of the sphere that contains the Rotunda in such a way that all the vertices of the Rotunda are touching the sphere. It is an important geometric property of this polyhedral shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{1}{2} \times (1 + \sqrt{5}) \times \sqrt{\frac{TSA}{\frac{1}{2} \times ((5 \times \sqrt{3}) + \sqrt{10 \times (65 + (29 \times \sqrt{5}))})}} \]

Where:

\( r_c \) — Circumsphere Radius of Rotunda (m)
\( TSA \) — Total Surface Area of Rotunda (m²)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the circumsphere radius based on the total surface area of the rotunda, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions of the rotunda, its relationship to circumscribed spheres, and for various applications in geometry, architecture, and 3D modeling.

4. Using the Calculator

Tips: Enter the total surface area of the rotunda in square meters. The value must be positive and valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rotunda in geometry?
A: A rotunda is a polyhedron that consists of a pentagonal base, a decagonal midsection, and a pentagonal top, forming a specific geometric shape with particular symmetry properties.

Q2: How is the circumsphere radius different from other radii?
A: The circumsphere radius specifically refers to the radius of the sphere that passes through all vertices of the polyhedron, unlike insphere radius which touches all faces or midsphere radius which touches all edges.

Q3: What are typical values for circumsphere radius?
A: The values depend on the size of the rotunda. For a standard rotunda with unit edge length, the circumsphere radius has a specific constant value derived from its geometric properties.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the rotunda polyhedron and uses constants and relationships unique to this particular geometric shape.

Q5: What practical applications does this calculation have?
A: This calculation is useful in architectural design, 3D modeling, crystallography, and any field dealing with polyhedral structures and their spatial properties.