Total Surface Area of Regular Bipyramid given Total Height Calculator

Formula Used:

\[ TSA = n \times l_e(Base) \times \sqrt{\left(\frac{h_{Total}}{2}\right)^2 + \left(\frac{1}{4} \times l_e(Base)^2 \times \left(\cot\left(\frac{\pi}{n}\right)\right)^2\right)} \]

Number of Base Vertices (n):

vertices

Edge Length of Base (lₑ):

Total Height (hₜₒₜₐₗ):

Total Surface Area (TSA):

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1. What is Total Surface Area of Regular Bipyramid?

The Total Surface Area of a Regular Bipyramid is the total amount of two-dimensional space occupied by all the faces of the Regular Bipyramid. It consists of the combined surface areas of both pyramid halves.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = n \times l_e(Base) \times \sqrt{\left(\frac{h_{Total}}{2}\right)^2 + \left(\frac{1}{4} \times l_e(Base)^2 \times \left(\cot\left(\frac{\pi}{n}\right)\right)^2\right)} \]

Where:

\( n \) — Number of base vertices
\( l_e(Base) \) — Edge length of base (m)
\( h_{Total} \) — Total height of bipyramid (m)
\( \pi \) — Mathematical constant pi (≈3.14159)
\( \cot \) — Cotangent trigonometric function

Explanation: The formula calculates the total surface area by considering the base perimeter and the lateral surface areas of both pyramid halves.

3. Importance of TSA Calculation

Details: Calculating total surface area is crucial for determining material requirements, heat transfer calculations, and understanding the geometric properties of bipyramidal structures in various engineering and architectural applications.

4. Using the Calculator

Tips: Enter the number of base vertices (minimum 3), base edge length in meters, and total height in meters. 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 joining two identical pyramids base-to-base. All base edges are equal, and all lateral edges are equal.

Q2: Why is the cotangent function used in the formula?
A: The cotangent function helps calculate the apothem of the base polygon, which is essential for determining the lateral surface area of the pyramid halves.

Q3: What are common applications of bipyramids?
A: Bipyramidal structures are used in crystallography, architecture, molecular geometry, and various engineering designs.

Q4: How does the number of vertices affect the surface area?
A: More vertices generally increase the surface area as it creates more triangular faces, though the relationship is non-linear due to the trigonometric components.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit, but you can use any consistent unit system as long as all length measurements are in the same units.