Half Height of Regular Bipyramid given Total Surface Area Calculator

Formula Used:

\[ h_{Half} = \sqrt{\left(\frac{TSA}{l_e \cdot n}\right)^2 - \left(\frac{1}{4} \cdot l_e^2 \cdot \left(\cot\left(\frac{\pi}{n}\right)\right)^2\right)} \]

Half Height of Regular Bipyramid (h_Half):

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1. What is the Half Height of Regular Bipyramid?

The Half Height of Regular Bipyramid is the total length of the perpendicular from the apex to the base of any one of the pyramids in the Regular Bipyramid. It represents half of the total height of the bipyramid.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h_{Half} = \sqrt{\left(\frac{TSA}{l_e \cdot n}\right)^2 - \left(\frac{1}{4} \cdot l_e^2 \cdot \left(\cot\left(\frac{\pi}{n}\right)\right)^2\right)} \]

Where:

\( h_{Half} \) — Half Height of Regular Bipyramid (m)
\( TSA \) — Total Surface Area of Regular Bipyramid (m²)
\( l_e \) — Edge Length of Base of Regular Bipyramid (m)
\( n \) — Number of Base Vertices of Regular Bipyramid
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \cot \) — Cotangent trigonometric function

Explanation: The formula calculates the half height based on the total surface area, base edge length, and number of base vertices, using trigonometric relationships.

3. Importance of Half Height Calculation

Details: Calculating the half height is essential for understanding the geometry and proportions of regular bipyramids, which is important in various fields including architecture, crystallography, and 3D modeling.

4. Using the Calculator

Tips: Enter total surface area in square meters, base edge length in meters, and number of base vertices (must be at least 3). All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a Regular Bipyramid?
A: A Regular Bipyramid is a polyhedron formed by two identical pyramids joined base-to-base, where the base is a regular polygon and the lateral faces are congruent isosceles triangles.

Q2: Why is the cotangent function used in the formula?
A: The cotangent function is used to relate the geometry of the regular polygonal base to the triangular faces of the bipyramid.

Q3: What is the minimum number of base vertices required?
A: The minimum number of base vertices is 3, which corresponds to a triangular bipyramid.

Q4: Can this calculator handle very large numbers?
A: The calculator can handle reasonably large numbers, but extremely large values may cause computational limitations.

Q5: How accurate are the results?
A: The results are accurate to 6 decimal places, which is sufficient for most practical applications involving geometric calculations.