Total Surface Area of Rhombicosidodecahedron given Circumsphere Radius Calculator

Formula Used:

\[ TSA = (30+(5\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times \left( \frac{2r_c}{\sqrt{11+(4\sqrt{5})}} \right)^2 \]

Total Surface Area of Rhombicosidodecahedron:

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1. What is Total Surface Area of Rhombicosidodecahedron?

The Total Surface Area of a Rhombicosidodecahedron is the total quantity of plane enclosed by the entire surface of this Archimedean solid. It consists of 20 regular triangular faces, 30 square faces, and 12 regular pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = (30+(5\sqrt{3})+(3\sqrt{25+(10\sqrt{5})})) \times \left( \frac{2r_c}{\sqrt{11+(4\sqrt{5})}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Rhombicosidodecahedron (m²)
\( r_c \) — Circumsphere Radius of Rhombicosidodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area based on the circumsphere radius, incorporating mathematical constants and geometric relationships specific to the rhombicosidodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, materials science, and 3D modeling. It helps in determining material requirements, heat transfer properties, and structural characteristics.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the total surface area using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombicosidodecahedron?
A: A rhombicosidodecahedron is an Archimedean solid with 62 faces: 20 triangles, 30 squares, and 12 pentagons. It has 60 vertices and 120 edges.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the rhombicosidodecahedron such that all vertices lie on the sphere's surface.

Q3: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before input, then convert the result back to your desired unit.

Q4: How accurate is the calculation?
A: The calculation uses precise mathematical constants and provides results accurate to 6 decimal places, making it suitable for most engineering and mathematical applications.

Q5: What are practical applications of this calculation?
A: This calculation is useful in architectural design, 3D printing, material science, and any field requiring precise geometric measurements of complex polyhedra.