Face Area of Dodecahedron given Insphere Radius Calculator

Formula Used:

\[ A_{Face} = \frac{1}{4} \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{2 \times r_i}{\sqrt{\frac{25 + (11 \times \sqrt{5})}{10}}} \right)^2 \]

Face Area of Dodecahedron:

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1. What is the Face Area of Dodecahedron given Insphere Radius?

The Face Area of Dodecahedron given Insphere Radius formula calculates the area of one of the 12 pentagonal faces of a regular dodecahedron when the radius of its inscribed sphere (insphere) is known. A dodecahedron is a polyhedron with 12 regular pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A_{Face} = \frac{1}{4} \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{2 \times r_i}{\sqrt{\frac{25 + (11 \times \sqrt{5})}{10}}} \right)^2 \]

Where:

\( A_{Face} \) — Face Area of Dodecahedron (m²)
\( r_i \) — Insphere Radius of Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula derives from the geometric relationship between the insphere radius and the face area of a regular dodecahedron, incorporating mathematical constants related to pentagonal geometry.

3. Importance of Face Area Calculation

Details: Calculating the face area of a dodecahedron is essential in geometry, architecture, material science, and 3D modeling. It helps in determining surface properties, material requirements, and structural characteristics of dodecahedral shapes.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the area of one pentagonal face of the dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular dodecahedron?
A: A regular dodecahedron is a three-dimensional shape with 12 identical regular pentagonal faces, 20 vertices, and 30 edges. It is one of the five Platonic solids.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the dodecahedron, touching all its faces at exactly one point each.

Q3: How is this formula derived?
A: The formula is derived from the geometric relationships between the insphere radius, edge length, and face area of a regular dodecahedron using trigonometric and algebraic methods.

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

Q5: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, game development, and any field dealing with regular polyhedral structures.