Circumsphere Radius of Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ r_c = \frac{\sqrt{3} \times (1+\sqrt{5}) \times r_m}{3+\sqrt{5}} \]

Circumsphere Radius of Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Circumsphere Radius of Dodecahedron?

The Circumsphere Radius of a Dodecahedron is the radius of the sphere that contains the Dodecahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{3} \times (1+\sqrt{5}) \times r_m}{3+\sqrt{5}} \]

Where:

\( r_c \) — Circumsphere Radius of Dodecahedron (m)
\( r_m \) — Midsphere Radius of Dodecahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula relates the circumsphere radius to the midsphere radius of a regular dodecahedron using mathematical constants derived from its geometric properties.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial for understanding the spatial dimensions of a dodecahedron, which has applications in crystallography, architecture, and various fields of mathematics and physics.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will automatically compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a three-dimensional shape with twelve flat faces, each being a regular pentagon. It is one of the five Platonic solids.

Q2: What is the difference between circumsphere and midsphere?
A: The circumsphere passes through all vertices of the dodecahedron, while the midsphere is tangent to all edges of the dodecahedron.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula is specifically for regular dodecahedrons where all faces are identical regular pentagons.

Q4: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and in the study of geometric properties of polyhedra.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons, as it's derived from the geometric properties of the shape.