Volume of Rhombic Triacontahedron Given Insphere Radius Calculator

Formula Used:

\[ V = 4 \times \left( \frac{r_i}{\sqrt{\frac{5 + 2\sqrt{5}}{5}}} \right)^3 \times \sqrt{5 + 2\sqrt{5}} \]

Volume (V):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Volume of Rhombic Triacontahedron?

The volume of a Rhombic Triacontahedron represents the amount of three-dimensional space enclosed by its surface. It is a polyhedron with 30 congruent rhombic faces, 32 vertices, and 60 edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = 4 \times \left( \frac{r_i}{\sqrt{\frac{5 + 2\sqrt{5}}{5}}} \right)^3 \times \sqrt{5 + 2\sqrt{5}} \]

Where:

\( V \) — Volume of Rhombic Triacontahedron (m³)
\( r_i \) — Insphere Radius (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the volume based on the insphere radius, which is the radius of the sphere that touches all faces of the polyhedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric shapes is fundamental in mathematics, engineering, and various scientific applications. For polyhedra like the Rhombic Triacontahedron, volume calculations help in understanding spatial properties and material requirements.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and non-zero. The calculator will compute the volume using the derived formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Rhombic Triacontahedron?
A: A Rhombic Triacontahedron is a convex polyhedron with 30 rhombic faces. It is one of the Catalan solids and is the dual polyhedron of the icosidodecahedron.

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: 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.

Q4: What are the applications of this calculation?
A: Volume calculations of polyhedra are used in crystallography, architecture, game development, and various engineering fields.

Q5: How accurate is the calculation?
A: The calculation uses precise mathematical formulas and provides results accurate to six decimal places.