Total Surface Area of Rhombic Dodecahedron given Insphere Radius Calculator

Formula Used:

\[ TSA = 12 \times \sqrt{2} \times r_i^2 \]

Total Surface Area of Rhombic Dodecahedron:

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1. What is the Total Surface Area of Rhombic Dodecahedron?

The Total Surface Area of a Rhombic Dodecahedron is the total area of all its twelve rhombic faces. It represents the complete outer surface measurement of this three-dimensional polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 12 \times \sqrt{2} \times r_i^2 \]

Where:

\( TSA \) — Total Surface Area of Rhombic Dodecahedron (m²)
\( r_i \) — Insphere Radius of Rhombic Dodecahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.414214)

Explanation: This formula calculates the total surface area based on the radius of the inscribed sphere (insphere) that touches all faces of the rhombic dodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, heat transfer calculations, coating requirements, and geometric analysis of this specific polyhedral structure.

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 in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombic Dodecahedron?
A: A rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It is a Catalan solid and the dual polyhedron of the cuboctahedron.

Q2: What does the insphere radius represent?
A: The insphere radius is the radius of the largest sphere that can be contained within the rhombic dodecahedron, touching all its faces.

Q3: Can this formula be used for any rhombic dodecahedron?
A: Yes, this formula applies to all regular rhombic dodecahedrons where all faces are congruent rhombi.

Q4: What are practical applications of this calculation?
A: This calculation is used in crystallography, material science, architecture, and any field dealing with this specific geometric shape.

Q5: How accurate is this formula?
A: The formula is mathematically exact for perfect rhombic dodecahedrons and provides precise surface area calculations.