Circumsphere Radius Of Icosidodecahedron Given Volume Calculator

Formula Used:

\[ r_c = \frac{1 + \sqrt{5}}{2} \times \left( \frac{6 \times V}{45 + 17 \times \sqrt{5}} \right)^{\frac{1}{3}} \]

Circumsphere Radius of Icosidodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Circumsphere Radius of Icosidodecahedron?

The Circumsphere Radius of an Icosidodecahedron is the radius of the sphere that contains the Icosidodecahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property of this Archimedean solid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{1 + \sqrt{5}}{2} \times \left( \frac{6 \times V}{45 + 17 \times \sqrt{5}} \right)^{\frac{1}{3}} \]

Where:

\( r_c \) — Circumsphere Radius of Icosidodecahedron (meters)
\( V \) — Volume of Icosidodecahedron (cubic meters)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula derives from the geometric properties of the icosidodecahedron and the relationship between its volume and circumscribed sphere radius.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions of the icosidodecahedron, its relationship with circumscribed spheres, and applications in geometry, crystallography, and architectural design.

4. Using the Calculator

Tips: Enter the volume of the icosidodecahedron in cubic meters. The volume must be a positive value greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosidodecahedron?
A: An icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 30 vertices, and 60 edges.

Q2: How is the circumsphere radius different from the insphere radius?
A: The circumsphere radius touches all vertices of the polyhedron, while the insphere radius is tangent to all faces.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometry research, molecular modeling, architectural design, and computer graphics.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the icosidodecahedron. Other polyhedra have different formulas for their circumsphere radii.

Q5: What is the golden ratio's role in this formula?
A: The term (1 + √5)/2 represents the golden ratio (φ ≈ 1.618), which appears frequently in the geometry of icosidodecahedra and other polyhedra with pentagonal symmetry.