Lateral Surface Area of Dodecahedron given Circumsphere Radius Calculator

Formula Used:

\[ LSA = \frac{5}{2} \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^2 \]

Lateral Surface Area of Dodecahedron:

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1. What is Lateral Surface Area of Dodecahedron?

The Lateral Surface Area of a Dodecahedron is the quantity of plane enclosed by all the lateral surfaces (excluding the top and bottom faces) of the Dodecahedron. A dodecahedron is a three-dimensional shape with 12 regular pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ LSA = \frac{5}{2} \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^2 \]

Where:

\( LSA \) — Lateral Surface Area of Dodecahedron (m²)
\( r_c \) — Circumsphere Radius of Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the lateral surface area based on the circumsphere radius, using mathematical constants and geometric relationships specific to a regular dodecahedron.

3. Importance of Lateral Surface Area Calculation

Details: Calculating the lateral surface area is important in geometry, architecture, and engineering for determining material requirements, surface coverage, and understanding the spatial properties of dodecahedral structures.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the lateral surface area in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a polyhedron with 12 flat faces, each being a regular pentagon. It is one of the five Platonic solids.

Q2: What is circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the dodecahedron such that all vertices lie on the sphere's surface.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular dodecahedron with the given circumsphere radius.

Q4: Can this calculator be used for irregular dodecahedrons?
A: No, this calculator is specifically designed for regular dodecahedrons where all edges and angles are equal.

Q5: What are practical applications of this calculation?
A: This calculation is useful in architectural design, 3D modeling, material science, and any field dealing with geometric properties of dodecahedral structures.