Total Surface Area Of Deltoidal Icositetrahedron Given Insphere Radius Calculator

Formula Used:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{r_i}{\sqrt{\frac{22 + 15\sqrt{2}}{34}}} \right)^2 \]

Total Surface Area of Deltoidal Icositetrahedron:

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1. What is the Total Surface Area of Deltoidal Icositetrahedron?

The Deltoidal Icositetrahedron is a Catalan solid with 24 deltoid faces. Its total surface area represents the sum of the areas of all its faces, providing a measure of the external coverage of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{12}{7} \times \sqrt{61 + 38\sqrt{2}} \times \left( \frac{r_i}{\sqrt{\frac{22 + 15\sqrt{2}}{34}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Deltoidal Icositetrahedron (m²)
\( r_i \) — Insphere Radius of Deltoidal Icositetrahedron (m)

Explanation: This formula calculates the total surface area based on the insphere radius, utilizing geometric properties and mathematical constants specific to the Deltoidal Icositetrahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is crucial for various applications including material estimation, heat transfer analysis, and understanding the geometric properties of this particular polyhedron in mathematical and engineering contexts.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-zero. The calculator will compute the corresponding total surface area of the Deltoidal Icositetrahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Icositetrahedron?
A: It's a Catalan solid with 24 faces, each of which is a deltoid (kite-shaped quadrilateral). It is the dual of the rhombicuboctahedron.

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

Q3: Are there other ways to calculate the surface area?
A: Yes, the surface area can also be calculated using edge length or other geometric parameters, but this calculator specifically uses the insphere radius.

Q4: What are typical values for the insphere radius?
A: The insphere radius depends on the size of the polyhedron. For a standard-sized Deltoidal Icositetrahedron, it typically ranges from centimeters to meters depending on the application.

Q5: Can this calculator handle very large or very small values?
A: Yes, the calculator can handle a wide range of positive values, though extremely large values may be limited by computational precision.