Circumsphere Radius of Icosahedron given Face Area Calculator

Formula Used:

\[ r_c = \frac{\sqrt{10 + (2 \times \sqrt{5})}}{4} \times \sqrt{\frac{4 \times A_{\text{Face}}}{\sqrt{3}}} \]

Circumsphere Radius of Icosahedron (r_c):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Circumsphere Radius of Icosahedron?

The Circumsphere Radius of an Icosahedron is the radius of the sphere that contains the icosahedron in such a way that all the vertices are lying on the sphere. It's a fundamental geometric property of this regular polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{\sqrt{10 + (2 \times \sqrt{5})}}{4} \times \sqrt{\frac{4 \times A_{\text{Face}}}{\sqrt{3}}} \]

Where:

\( r_c \) — Circumsphere Radius of Icosahedron (m)
\( A_{\text{Face}} \) — Face Area of Icosahedron (m²)

Explanation: The formula derives from the geometric properties of regular icosahedrons, relating the circumsphere radius to the face area through mathematical constants and relationships.

3. Importance of Circumsphere Radius Calculation

Details: Calculating the circumsphere radius is crucial in geometry, 3D modeling, and various engineering applications where understanding the spatial dimensions and relationships of icosahedral structures is important.

4. Using the Calculator

Tips: Enter the face area of the icosahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is an icosahedron?
A: An icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 12 vertices, and 30 edges.

Q2: How is face area related to circumsphere radius?
A: The face area determines the size of the icosahedron, which in turn determines the radius of its circumscribing sphere through the mathematical relationship shown in the formula.

Q3: What are typical applications of this calculation?
A: This calculation is used in molecular modeling, architectural design, game development, and any field dealing with three-dimensional geometric structures.

Q4: Can this formula be used for irregular icosahedrons?
A: No, this formula applies only to regular icosahedrons where all faces are identical equilateral triangles.

Q5: What units should be used for input?
A: The calculator expects face area in square meters, but any consistent area unit can be used as long as the output radius is interpreted in the corresponding length unit.