Edge Length of Square Cupola given Height Calculator

Formula Used:

\[ l_e = \frac{h}{\sqrt{1 - \frac{1}{4} \csc\left(\frac{\pi}{4}\right)^2}} \]

Edge Length of Square Cupola:

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1. What is the Edge Length of Square Cupola?

The Edge Length of Square Cupola is the length of any edge of the Square Cupola, a polyhedron with one square face, four triangular faces, and one octagonal face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \frac{h}{\sqrt{1 - \frac{1}{4} \csc\left(\frac{\pi}{4}\right)^2}} \]

Where:

\( l_e \) — Edge Length of Square Cupola (m)
\( h \) — Height of Square Cupola (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \csc \) — Cosecant trigonometric function

Explanation: This formula calculates the edge length based on the height of the square cupola using trigonometric relationships.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for geometric analysis, architectural design, and understanding the properties of polyhedral structures.

4. Using the Calculator

Tips: Enter the height of the square cupola in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Square Cupola?
A: A Square Cupola is a polyhedron with one square face, four triangular faces, and one octagonal face, forming a cupola-like structure.

Q2: What are the units for measurement?
A: Both height and edge length are measured in meters (m) in this calculator.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula is derived for the Square Cupola geometry only.

Q4: What is the range of valid input values?
A: The height must be a positive real number greater than zero.

Q5: How accurate is the calculation?
A: The calculation uses double-precision floating-point arithmetic and provides results accurate to 6 decimal places.