Volume Of Elongated Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = 6 \times \left( \frac{TSA}{2\sqrt{3}(3+\sqrt{5})} \right)^{\frac{3}{2}} \]

Volume of Elongated Dodecahedron:

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1. What is Volume of Elongated Dodecahedron?

The Volume of Elongated Dodecahedron is the total quantity of three dimensional space enclosed by the surface of the Elongated Dodecahedron. It's a geometric measurement that represents the capacity of this particular polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ V = 6 \times \left( \frac{TSA}{2\sqrt{3}(3+\sqrt{5})} \right)^{\frac{3}{2}} \]

Where:

\( V \) — Volume of Elongated Dodecahedron (m³)
\( TSA \) — Total Surface Area of Elongated Dodecahedron (m²)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula derives the volume from the total surface area using geometric relationships specific to the elongated dodecahedron shape.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in mathematics, engineering, architecture, and various scientific fields. It helps in understanding spatial properties, material requirements, and structural characteristics.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is an Elongated Dodecahedron?
A: An elongated dodecahedron is a polyhedron formed by elongating a regular dodecahedron along one of its axes, creating a shape with 12 pentagonal faces and additional rectangular faces.

Q2: Why is the formula so complex?
A: The formula complexity arises from the geometric relationships between surface area and volume in this specific polyhedron, involving square roots and fractional exponents.

Q3: What units should I use?
A: Use consistent units - typically square meters for surface area and cubic meters for volume. The calculator maintains this consistency.

Q4: Can this calculator handle very large or small values?
A: Yes, within reasonable computational limits. Extremely large values might cause precision issues due to floating-point arithmetic limitations.

Q5: Is this formula applicable to all dodecahedron variations?
A: No, this specific formula applies only to the elongated dodecahedron. Other dodecahedron variations have different volume formulas.