Height Of Antiprism Given Total Surface Area Calculator

Formula Used:

\[ h = \sqrt{1-\frac{(\sec(\frac{\pi}{2 \times N}))^2}{4}} \times \sqrt{\frac{TSA}{\frac{N}{2} \times (\cot(\frac{\pi}{N}) + \sqrt{3})}} \]

Height of Antiprism (h):

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1. What is the Height of Antiprism Calculation?

The height of an antiprism is the vertical distance between its two parallel polygonal faces. This calculator determines the height based on the total surface area and number of vertices using a specialized geometric formula.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ h = \sqrt{1-\frac{(\sec(\frac{\pi}{2 \times N}))^2}{4}} \times \sqrt{\frac{TSA}{\frac{N}{2} \times (\cot(\frac{\pi}{N}) + \sqrt{3})}} \]

Where:

\( h \) — Height of Antiprism (m)
\( N \) — Number of Vertices of Antiprism
\( TSA \) — Total Surface Area of Antiprism (m²)
\( \pi \) — Mathematical constant pi (≈3.14159)

Explanation: The formula combines trigonometric functions (secant and cotangent) with square roots to calculate the height based on the geometric properties of antiprisms.

3. Importance of Height Calculation

Details: Calculating the height of an antiprism is essential for understanding its three-dimensional geometry, volume calculation, and for applications in crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the number of vertices (must be ≥3) and the total surface area (must be >0). The calculator will compute the height using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is an antiprism?
A: An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles.

Q2: Why are trigonometric functions used in this calculation?
A: Trigonometric functions help describe the angular relationships and proportions between the different faces and edges of the antiprism.

Q3: What is the minimum number of vertices required?
A: The minimum number of vertices is 3, which creates a triangular antiprism (the simplest form).

Q4: Can this formula be used for all types of antiprisms?
A: Yes, this formula works for regular antiprisms with any number of vertices ≥3.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for ideal regular antiprisms, assuming precise input values.