Pyramidal Height Of Small Stellated Dodecahedron Given Circumradius Calculator

Formula Used:

\[ h_{Pyramid} = \frac{\sqrt{25+10\sqrt{5}}}{5} \times \frac{4r_c}{\sqrt{50+22\sqrt{5}}} \]

Pyramidal Height of Small Stellated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Pyramidal Height of Small Stellated Dodecahedron?

The Pyramidal Height of Small Stellated Dodecahedron is the height of any of the inwards directed tetrahedral pyramids of the Small Stellated Dodecahedron. It represents the perpendicular distance from the center of the base pyramid to its apex.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h_{Pyramid} = \frac{\sqrt{25+10\sqrt{5}}}{5} \times \frac{4r_c}{\sqrt{50+22\sqrt{5}}} \]

Where:

\( h_{Pyramid} \) — Pyramidal Height of Small Stellated Dodecahedron (m)
\( r_c \) — Circumradius of Small Stellated Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the pyramidal height based on the circumradius of the Small Stellated Dodecahedron, using mathematical constants derived from its geometric properties.

3. Importance of Pyramidal Height Calculation

Details: Calculating the pyramidal height is essential for understanding the three-dimensional structure of the Small Stellated Dodecahedron, determining its volume, surface area, and other geometric properties.

4. Using the Calculator

Tips: Enter the circumradius of the Small Stellated Dodecahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron that consists of 12 pentagrammic faces with five pentagrams meeting at each vertex.

Q2: How is circumradius related to pyramidal height?
A: The circumradius determines the overall size of the polyhedron, which in turn affects the pyramidal height through the mathematical relationship expressed in the formula.

Q3: What are the applications of this calculation?
A: This calculation is used in geometry, mathematical modeling, architectural design, and in understanding the properties of complex polyhedra.

Q4: Are there limitations to this formula?
A: This formula is specifically designed for the Small Stellated Dodecahedron and may not apply to other polyhedra or geometric shapes.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most mathematical and engineering applications.