Volume Of Icosidodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \frac{45 + 17\sqrt{5}}{6} \times \left( \sqrt{ \frac{TSA}{(5\sqrt{3}) + (3\sqrt{25 + 10\sqrt{5}})} } \right)^3 \]

Volume of Icosidodecahedron:

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1. What is the Volume of Icosidodecahedron?

The volume of an icosidodecahedron represents the total three-dimensional space enclosed by its surface. An icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 60 edges, and 30 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{45 + 17\sqrt{5}}{6} \times \left( \sqrt{ \frac{TSA}{(5\sqrt{3}) + (3\sqrt{25 + 10\sqrt{5}})} } \right)^3 \]

Where:

\( V \) — Volume of the icosidodecahedron
\( TSA \) — Total surface area of the icosidodecahedron
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula derives the volume from the total surface area by first calculating the edge length and then applying the standard volume formula for an icosidodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields. It helps in material estimation, structural analysis, and understanding spatial properties of three-dimensional objects.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is an icosidodecahedron?
A: An icosidodecahedron is a semi-regular polyhedron with 32 faces (20 equilateral triangles and 12 regular pentagons), 60 edges, and 30 vertices.

Q2: What are the applications of this calculation?
A: This calculation is used in geometry, architectural design, material science, and 3D modeling where precise volume measurements of complex polyhedra are required.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is designed only for the icosidodecahedron. Other polyhedra have different volume formulas based on their unique geometric properties.

Q4: What if I have the edge length instead of surface area?
A: If you have the edge length (a), you can use the simpler volume formula: \( V = \frac{45 + 17\sqrt{5}}{6} \times a^3 \)

Q5: How accurate is this calculation?
A: The calculation is mathematically precise when using exact values. The calculator provides results rounded to 6 decimal places for practical use.