Total Surface Area Of Elongated Dodecahedron Given Width Calculator

Formula Used:

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

Total Surface Area (TSA):

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1. What is the Total Surface Area of Elongated Dodecahedron?

The Total Surface Area of an Elongated Dodecahedron refers to the total quantity of plane enclosed on the entire surface of this three-dimensional geometric shape. It is an important measurement in geometry and material science.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area (m²)
\( w \) — Width of Elongated Dodecahedron (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 width measurement of the elongated dodecahedron, incorporating mathematical constants and geometric relationships specific to this shape.

3. Importance of TSA Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, structural analysis, packaging design, and understanding the geometric properties of this specific polyhedron.

4. Using the Calculator

Tips: Enter the width of the elongated dodecahedron in meters. The value must be positive and greater than zero. The calculator will compute the total surface area based on the geometric formula.

5. Frequently Asked Questions (FAQ)

Q1: What is an elongated dodecahedron?
A: An elongated dodecahedron is a polyhedron created by elongating a regular dodecahedron by inserting a cube between its two halves, resulting in a shape with 12 regular pentagonal faces and additional rectangular faces.

Q2: Why is the formula so complex?
A: The formula incorporates mathematical constants (√3 and √5) that arise from the geometric relationships and angles inherent in the dodecahedron's structure, particularly the golden ratio relationships.

Q3: What are the units for the result?
A: The result is given in square meters (m²), which is the standard unit for surface area measurements.

Q4: Can this calculator handle different units?
A: The calculator expects input in meters. If you have measurements in other units, convert them to meters first before using the calculator.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact based on the geometric formula. The result is rounded to 6 decimal places for practical use while maintaining high precision.