Total Surface Area of Great Icosahedron given Mid Ridge Length Calculator

Formula Used:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{2 \times l_{Ridge(Mid)}}{1 + \sqrt{5}} \right)^2 \]

Total Surface Area of Great Icosahedron:

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1. What is the Total Surface Area of Great Icosahedron?

The Total Surface Area of a Great Icosahedron refers to the total area of all the triangular faces that make up this complex polyhedron. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 3 \times \sqrt{3} \times (5 + 4 \times \sqrt{5}) \times \left( \frac{2 \times l_{Ridge(Mid)}}{1 + \sqrt{5}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Great Icosahedron (m²)
\( l_{Ridge(Mid)} \) — Mid Ridge Length of Great 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 based on the mid ridge length of the Great Icosahedron, incorporating mathematical constants and geometric relationships.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric shapes is fundamental in mathematics, physics, engineering, and architecture. It helps in understanding material requirements, heat transfer properties, and structural characteristics of three-dimensional objects.

4. Using the Calculator

Tips: Enter the mid ridge length in meters. The value must be positive and greater than zero. The calculator will compute the total surface area using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: A Great Icosahedron is a non-convex polyhedron with 20 triangular faces. It is one of the Kepler-Poinsot solids and has a complex star-like structure.

Q2: How is the mid ridge length defined?
A: The mid ridge length is the length of any of the edges that starts from the peak vertex and ends on the interior of the pentagon on which each peak of the Great Icosahedron is attached.

Q3: What units should I use for input?
A: The calculator accepts input in meters, and the result is given in square meters. You can convert from other units as needed before input.

Q4: How accurate is the calculation?
A: The calculation uses precise mathematical constants and formulas, providing results accurate to six decimal places.

Q5: Can this calculator handle very large or small values?
A: The calculator can handle a wide range of positive values, though extremely large or small numbers may be limited by PHP's floating-point precision.