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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 is a key geometric property used in various mathematical and engineering applications.
The calculator uses the formula:
Where:
Explanation: The formula calculates the circumsphere radius based on the given lateral surface area of the icosahedron, incorporating mathematical constants specific to this regular polyhedron.
Details: Calculating the circumsphere radius is essential for understanding the spatial dimensions of an icosahedron, which has applications in geometry, crystallography, and structural engineering.
Tips: Enter the lateral surface area in square meters. The value must be positive and greater than zero for accurate calculation.
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 Lateral Surface Area different from Total Surface Area?
A: Lateral Surface Area excludes the base areas (if any), but in a regular icosahedron, since all faces are identical, LSA typically refers to the total surface area of all faces.
Q3: What are the units for Circumsphere Radius?
A: The circumsphere radius is measured in meters (m), consistent with the input unit for lateral surface area (m²).
Q4: Can this formula be used for irregular icosahedrons?
A: No, this formula is specifically derived for regular icosahedrons where all faces are equilateral triangles and all vertices lie on a sphere.
Q5: What is the typical range of values for Circumsphere Radius?
A: The circumsphere radius depends on the size of the icosahedron. For standard mathematical problems, values can range from very small (mm scale) to very large (m scale or beyond).