Formula Used:
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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.
The calculator uses the formula:
Where:
Explanation: This formula calculates the total surface area based on the leg length of the isosceles triangular faces of the Pentakis Dodecahedron.
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.
Tips: Enter the leg length of the Pentakis Dodecahedron in meters. The value must be positive and greater than zero for accurate calculation.
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.