Total Surface Area of Icosidodecahedron given Midsphere Radius Calculator

Formula Used:

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

Total Surface Area (TSA):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Icosidodecahedron?

The Icosidodecahedron is an Archimedean solid with 32 faces (20 triangles and 12 pentagons), 60 edges, and 30 vertices. It represents one of the 13 convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( TSA \) — Total Surface Area of the Icosidodecahedron
\( r_m \) — Midsphere Radius of the Icosidodecahedron
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the total surface area of an icosidodecahedron based on its midsphere radius, incorporating the mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Surface Area Calculation

Details: Calculating the total surface area of geometric solids is fundamental in various fields including architecture, materials science, and 3D modeling. For the icosidodecahedron, this calculation helps in understanding its geometric properties and applications in design and construction.

4. Using the Calculator

Tips: Enter the midsphere radius value 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 midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q2: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before calculation.

Q3: What is the precision of the calculation?
A: The result is rounded to 6 decimal places for practical use while maintaining mathematical accuracy.

Q4: Are there other ways to calculate surface area?
A: Yes, surface area can also be calculated using edge length or circumsphere radius, but this calculator specifically uses midsphere radius.

Q5: What applications use icosidodecahedrons?
A: Icosidodecahedrons are used in architectural design, molecular modeling, game development, and as geometric puzzles.