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Total Surface Area Of Pentakis Dodecahedron Given Leg Length Calculator

Formula Used:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162 \times \sqrt{5}} \times \left( \frac{38 \times lLeg}{3 \times (9 + \sqrt{5})} \right)^2 \]

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1. What Is The Total Surface Area Of Pentakis Dodecahedron Given Leg Length?

The Total Surface Area of a Pentakis Dodecahedron given its Leg Length refers to the total area covered by all the faces of this geometric solid. The Pentakis Dodecahedron is a Catalan solid derived from the dodecahedron by placing a pyramid on each face.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{19} \times \sqrt{413 + 162 \times \sqrt{5}} \times \left( \frac{38 \times lLeg}{3 \times (9 + \sqrt{5})} \right)^2 \]

Where:

Explanation: This formula calculates the total surface area based on the leg length of the isosceles triangular faces of the Pentakis Dodecahedron.

3. Importance Of Total Surface Area Calculation

Details: Calculating the total surface area is important in various fields including geometry, architecture, material science, and 3D modeling where precise measurements of polyhedral surfaces are required.

4. Using The Calculator

Tips: Enter the leg length of the Pentakis Dodecahedron in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid with 60 isosceles triangular faces, 90 edges, and 32 vertices, formed by placing a pyramid on each face of a regular dodecahedron.

Q2: How is this different from a regular dodecahedron?
A: While a regular dodecahedron has 12 pentagonal faces, the Pentakis Dodecahedron has 60 triangular faces and is the dual of the truncated icosahedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in architectural design, crystalography, geodesic dome construction, and mathematical modeling of complex polyhedra.

Q4: Can this formula be used for any Pentakis Dodecahedron?
A: Yes, this formula applies to all regular Pentakis Dodecahedra where all triangular faces are congruent isosceles triangles.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for ideal geometric shapes, though real-world measurements may have practical limitations.

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