Insphere Radius of Dodecahedron given Total Surface Area Calculator

Formula Used:

\[ r_i = \sqrt{\frac{TSA \times (25 + 11\sqrt{5})}{120 \times \sqrt{25 + 10\sqrt{5}}}} \]

Insphere Radius of Dodecahedron:

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1. What is the Insphere Radius of Dodecahedron?

The Insphere Radius of a Dodecahedron is the radius of the sphere that is contained by the Dodecahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \sqrt{\frac{TSA \times (25 + 11\sqrt{5})}{120 \times \sqrt{25 + 10\sqrt{5}}}} \]

Where:

\( r_i \) — Insphere Radius of Dodecahedron (m)
\( TSA \) — Total Surface Area of Dodecahedron (m²)
\( \sqrt{5} \) — Mathematical constant (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the total surface area of a regular dodecahedron, using the mathematical properties of this Platonic solid.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling for understanding the spatial relationships within a dodecahedron and for applications in crystallography, architecture, and molecular modeling.

4. Using the Calculator

Tips: Enter the total surface area of the dodecahedron in square meters. The value must be positive and greater than zero.

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: How is the insphere radius different from the circumsphere radius?
A: The insphere radius is the radius of the sphere that fits inside the dodecahedron and touches all faces, while the circumsphere radius is the radius of the sphere that contains the dodecahedron with all vertices touching the sphere.

Q3: What are the units for the insphere radius?
A: The insphere radius is measured in meters (m), the same unit as used for the input surface area.

Q4: Can this formula be used for irregular dodecahedrons?
A: No, this formula is specifically for regular dodecahedrons where all faces are identical regular pentagons.

Q5: What is the significance of the constants in the formula?
A: The constants (25, 11, 120, 10) are derived from the geometric properties of the regular dodecahedron and the mathematical constant φ (phi), the golden ratio.