Total Surface Area Of Hexakis Octahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{12}{7} \times \sqrt{543 + 176\sqrt{2}} \times (r_i)^2 \times \frac{1}{\frac{402 + 195\sqrt{2}}{194}} \]

Total Surface Area of Hexakis Octahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Total Surface Area of Hexakis Octahedron?

The Total Surface Area of a Hexakis Octahedron is the total area of all its faces. A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron, featuring 48 faces, 72 edges, and 26 vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{12}{7} \times \sqrt{543 + 176\sqrt{2}} \times (r_i)^2 \times \frac{1}{\frac{402 + 195\sqrt{2}}{194}} \]

Where:

\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( r_i \) — Insphere Radius of Hexakis Octahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the insphere radius, incorporating mathematical constants and geometric relationships specific to the Hexakis Octahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is essential in various fields including mathematics, engineering, architecture, and materials science. It helps in determining material requirements, heat transfer properties, and structural characteristics.

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 total surface area using the specialized formula for Hexakis Octahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid with 48 faces, 72 edges, and 26 vertices. It is the dual polyhedron of the truncated cuboctahedron.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, touching all its faces.

Q3: Why is the formula so complex?
A: The formula incorporates mathematical constants and relationships specific to the geometry of the Hexakis Octahedron, requiring square roots and precise coefficients for accurate calculation.

Q4: What are the practical applications of this calculation?
A: This calculation is useful in crystallography, architectural design, and materials engineering where precise geometric measurements are required.

Q5: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for Hexakis Octahedra. Other polyhedra require different formulas for surface area calculation.