Ridge Length of Star Pyramid given Lateral Edge Length Calculator

Formula Used:

\[ l_{ridge} = \sqrt{\left(\frac{l_{pentagon}^2}{100} \times (50 + 10\sqrt{5})\right) + l_{lateral}^2 - \left(\frac{l_c^2}{100} \times (50 + 10\sqrt{5})\right)} \]

Pentagonal Edge Length of Base of Star Pyramid:

Lateral Edge Length of Star Pyramid:

Chord Length of Star Pyramid:

Ridge Length of Star Pyramid:

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1. What is the Ridge Length of Star Pyramid?

The Ridge Length of Star Pyramid is the length of the line joining any inner vertex of base of the Star Pyramid and the apex of the Star Pyramid. It represents the diagonal distance from the base's inner vertex to the pyramid's peak.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_{ridge} = \sqrt{\left(\frac{l_{pentagon}^2}{100} \times (50 + 10\sqrt{5})\right) + l_{lateral}^2 - \left(\frac{l_c^2}{100} \times (50 + 10\sqrt{5})\right)} \]

Where:

\( l_{ridge} \) — Ridge Length of Star Pyramid (m)
\( l_{pentagon} \) — Pentagonal Edge Length of Base of Star Pyramid (m)
\( l_{lateral} \) — Lateral Edge Length of Star Pyramid (m)
\( l_c \) — Chord Length of Star Pyramid (m)

Explanation: This formula calculates the ridge length based on the geometric properties of the star pyramid, incorporating the golden ratio through the √5 term.

3. Importance of Ridge Length Calculation

Details: Calculating the ridge length is essential for understanding the three-dimensional geometry of star pyramids, which is important in architectural design, crystallography, and mathematical modeling of complex polyhedral structures.

4. Using the Calculator

Tips: Enter all measurements in meters. Ensure all values are positive numbers. The pentagonal edge length, lateral edge length, and chord length must all be greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Star Pyramid?
A: A Star Pyramid is a pyramid with a star-shaped (pentagrammic) base, typically constructed from a regular pentagon with isosceles triangles attached to form the star pattern.

Q2: How is the Ridge Length different from Lateral Edge Length?
A: The Lateral Edge Length connects outer vertices to the apex, while the Ridge Length connects inner vertices to the apex, creating different diagonal measurements.

Q3: What does the √5 term represent in the formula?
A: The √5 term relates to the golden ratio (φ = (1+√5)/2), which appears naturally in pentagonal and pentagrammic geometry due to their five-fold symmetry.

Q4: Can this formula be used for any star pyramid?
A: This specific formula applies to star pyramids with a regular pentagrammic base. Different base shapes would require different geometric relationships.

Q5: What are practical applications of this calculation?
A: This calculation is used in architectural design of star-shaped structures, mathematical research in geometry, and in understanding the properties of certain crystalline structures with pentagonal symmetry.