Volume of Antiprism given Surface to Volume Ratio Calculator

Volume of Antiprism Formula:

\[ V = \frac{N_{\text{Vertices}} \cdot \sin\left(\frac{3\pi}{2N_{\text{Vertices}}}\right) \cdot \sqrt{4\cos^2\left(\frac{\pi}{2N_{\text{Vertices}}}\right)-1} \cdot \left(\frac{6\sin^2\left(\frac{\pi}{N_{\text{Vertices}}}\right) \cdot (\cot\left(\frac{\pi}{N_{\text{Vertices}}}\right)+\sqrt{3})}{\sin\left(\frac{3\pi}{2N_{\text{Vertices}}}\right) \cdot \sqrt{4\cos^2\left(\frac{\pi}{2N_{\text{Vertices}}}\right)-1} \cdot R_{A/V}}\right)^3}{12\sin^2\left(\frac{\pi}{N_{\text{Vertices}}}\right)} \]

Volume of Antiprism:

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1. What is Volume of Antiprism?

The volume of an antiprism is defined as the quantity of three-dimensional space enclosed by its closed surface. An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles.

2. How Does the Calculator Work?

The calculator uses the complex formula:

\[ V = \frac{N \cdot \sin\left(\frac{3\pi}{2N}\right) \cdot \sqrt{4\cos^2\left(\frac{\pi}{2N}\right)-1} \cdot \left(\frac{6\sin^2\left(\frac{\pi}{N}\right) \cdot (\cot\left(\frac{\pi}{N}\right)+\sqrt{3})}{\sin\left(\frac{3\pi}{2N}\right) \cdot \sqrt{4\cos^2\left(\frac{\pi}{2N}\right)-1} \cdot R_{A/V}}\right)^3}{12\sin^2\left(\frac{\pi}{N}\right)} \]

Where:

\( V \) — Volume of Antiprism (m³)
\( N \) — Number of Vertices of Antiprism
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \pi \) — Archimedes' constant (≈ 3.14159)

Explanation: This complex formula combines trigonometric functions to calculate the volume based on the number of vertices and surface to volume ratio.

3. Importance of Volume Calculation

Details: Calculating the volume of antiprisms is important in geometry, crystallography, and materials science for understanding spatial properties and structural characteristics of these polyhedral shapes.

4. Using the Calculator

Tips: Enter the number of vertices (must be ≥3) and the surface to volume ratio. The calculator will compute the volume using the complex trigonometric formula.

5. Frequently Asked Questions (FAQ)

Q1: What is the minimum number of vertices required?
A: An antiprism must have at least 3 vertices to form a valid polyhedron.

Q2: What units are used for the result?
A: The volume is calculated in cubic meters (m³), assuming consistent units for all inputs.

Q3: Can this formula be used for all antiprisms?
A: This specific formula is designed for regular antiprisms where all edges have equal length and all faces are regular polygons.

Q4: What if I get an error in calculation?
A: Ensure that the number of vertices is at least 3 and the surface to volume ratio is a positive number.

Q5: How accurate is this calculation?
A: The calculation uses precise trigonometric functions and should be mathematically accurate for valid inputs.