Total Surface Area of Regular Bipyramid Calculator

Total Surface Area Formula:

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

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 sum of the areas of all its faces. A regular bipyramid consists of two identical pyramids joined base-to-base, with a regular polygon as the base.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula calculates the total surface area by considering the triangular faces of both pyramids in the bipyramid structure.

3. Importance of TSA Calculation

Details: Calculating the total surface area is essential for various applications including material estimation, structural design, heat transfer analysis, and geometric modeling of bipyramidal structures.

4. Using the Calculator

Tips: Enter the number of base vertices (minimum 3), edge length of base in meters, and half 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 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 helps calculate the apothem of the base polygon, which is essential for determining the area of the triangular faces.

Q3: What is the minimum number of base vertices required?
A: The minimum number is 3, which corresponds to a triangular bipyramid (a regular octahedron when equilateral).

Q4: How does half height differ from total height?
A: Half height refers to the height from the base to one apex. The total height of the bipyramid is twice the half height.

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.