Lateral Surface Area of Dodecahedron given Insphere Radius Calculator

Formula Used:

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

Lateral Surface Area of Dodecahedron:

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

The Lateral Surface Area of a Dodecahedron given its Insphere Radius is the area of all the lateral faces (excluding the top and bottom faces) of a regular dodecahedron, calculated using the radius of the sphere inscribed within it.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( LSA \) — Lateral Surface Area of Dodecahedron (m²)
\( r_i \) — Insphere Radius of Dodecahedron (m)

Explanation: This formula derives the lateral surface area from the insphere radius by incorporating geometric constants related to the dodecahedron's structure.

3. Importance of Lateral Surface Area Calculation

Details: Calculating the lateral surface area is essential in geometry and engineering for determining material requirements, heat transfer, and other surface-related properties of dodecahedral structures.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Dodecahedron?
A: A dodecahedron is a polyhedron with twelve flat faces, each being a regular pentagon.

Q2: What is Insphere Radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the dodecahedron, touching all its faces.

Q3: How accurate is this formula?
A: The formula is mathematically exact for a regular dodecahedron.

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 architecture, material science, and 3D modeling where dodecahedral shapes are employed.