Total Surface Area of Icosahedron given Insphere Radius Calculator

Formula Used:

\[ TSA = 5 \times \sqrt{3} \times \left( \frac{12 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \right)^2 \]

Total Surface Area of Icosahedron:

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1. What is Total Surface Area of Icosahedron?

The Total Surface Area of an Icosahedron is the total quantity of plane enclosed by the entire surface of this regular polyhedron, which has 20 equilateral triangular faces, 12 vertices, and 30 edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 5 \times \sqrt{3} \times \left( \frac{12 \times r_i}{\sqrt{3} \times (3 + \sqrt{5})} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Icosahedron (m²)
\( r_i \) — Insphere Radius of Icosahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the total surface area of a regular icosahedron based on the radius of its inscribed sphere (insphere).

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, engineering, material science, and 3D modeling for determining material requirements, heat transfer properties, and structural analysis.

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 total surface area of the icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is an Icosahedron?
A: An icosahedron is a regular polyhedron with 20 equilateral triangular faces, 12 vertices, and 30 edges. It is one of the five Platonic solids.

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

Q3: Can this formula be used for irregular icosahedrons?
A: No, this formula is specifically for regular icosahedrons where all faces are equilateral triangles and all vertices are equivalent.

Q4: What are the units of measurement?
A: The insphere radius should be in meters, and the resulting surface area will be in square meters. You can convert from other units as needed.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular icosahedron. The accuracy depends on the precision of the input value and the computational precision.