Leg Length Of Pentakis Dodecahedron Given Insphere Radius Calculator

Formula Used:

\[ l_{Leg} = \frac{3}{38} \times (9 + \sqrt{5}) \times \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \]

Leg Length (l_Leg):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Leg Length of Pentakis Dodecahedron?

The Leg Length of Pentakis Dodecahedron refers to the length of the equal sides of the isosceles triangular faces that make up this polyhedron. It is a key geometric parameter in understanding the structure and properties of the Pentakis Dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{Leg} = \frac{3}{38} \times (9 + \sqrt{5}) \times \frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}} \]

Where:

\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the leg length based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Pentakis Dodecahedron.

3. Importance of Leg Length Calculation

Details: Calculating the leg length is essential for understanding the geometry of the Pentakis Dodecahedron, including its surface area, volume, and other geometric properties. It is particularly useful in fields such as crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding leg length of the Pentakis Dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 60 isosceles triangular faces.

Q2: How is the insphere radius defined?
A: The insphere radius is the radius of the largest sphere that can fit inside the Pentakis Dodecahedron, touching all its faces.

Q3: What are typical values for the leg length?
A: The leg length depends on the size of the Pentakis Dodecahedron. For a given insphere radius, the leg length is determined by the specific geometric proportions of the solid.

Q4: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Pentakis Dodecahedron due to its unique geometric properties.

Q5: What units should be used?
A: The calculator uses meters for both input and output, but any consistent unit of length can be used as long as it is the same for both values.