Surface To Volume Ratio Of Hexakis Octahedron Given Total Surface Area Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{543+176\sqrt{2}}}{\sqrt{6(986+607\sqrt{2})}} \times \sqrt{\frac{3\sqrt{543+176\sqrt{2}}}{7 \times \text{Total Surface Area}}} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Hexakis Octahedron?

The Surface to Volume Ratio of a Hexakis Octahedron is a geometric property that represents the relationship between the total surface area and the volume of this particular polyhedron. It indicates how much surface area is available per unit volume of the shape.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{543+176\sqrt{2}}}{\sqrt{6(986+607\sqrt{2})}} \times \sqrt{\frac{3\sqrt{543+176\sqrt{2}}}{7 \times \text{Total Surface Area}}} \]

Where:

Total Surface Area (TSA) — The total surface area of the Hexakis Octahedron in square meters
\(\sqrt{}\) — Square root function
\(\sqrt{2}\) — Mathematical constant approximately equal to 1.4142

Explanation: This complex formula calculates the surface to volume ratio based on the geometric properties of the Hexakis Octahedron and its total surface area.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and engineering. It helps understand properties like heat transfer, chemical reactivity, and mechanical strength of geometric structures.

4. Using the Calculator

Tips: Enter the total surface area of the Hexakis Octahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: What units are used for the calculation?
A: The total surface area should be in square meters (m²), and the result will be in reciprocal meters (m⁻¹).

Q3: Why is the formula so complex?
A: The complexity arises from the intricate geometry of the Hexakis Octahedron, which requires precise mathematical relationships to calculate its properties accurately.

Q4: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of values, but extremely large or small numbers may be limited by PHP's floating-point precision.

Q5: What are typical values for surface to volume ratio?
A: The surface to volume ratio depends on the size of the Hexakis Octahedron. Smaller objects generally have higher surface to volume ratios than larger ones of the same shape.