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Volume of Antiprism Calculator

Volume of Antiprism Formula:

\[ V = \frac{N \cdot \sin\left(\frac{3\pi}{2N}\right) \cdot \sqrt{4 \cdot \cos^2\left(\frac{\pi}{2N}\right) - 1} \cdot l_e^3}{12 \cdot \sin^2\left(\frac{\pi}{N}\right)} \]

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1. What is the Volume of Antiprism?

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

2. How Does the Calculator Work?

The calculator uses the antiprism volume formula:

\[ V = \frac{N \cdot \sin\left(\frac{3\pi}{2N}\right) \cdot \sqrt{4 \cdot \cos^2\left(\frac{\pi}{2N}\right) - 1} \cdot l_e^3}{12 \cdot \sin^2\left(\frac{\pi}{N}\right)} \]

Where:

Explanation: The formula accounts for the geometric properties of antiprisms, using trigonometric functions to calculate the enclosed volume based on the number of vertices and edge length.

3. Importance of Volume Calculation

Details: Calculating the volume of antiprisms is crucial for geometric analysis, architectural design, and understanding the properties of polyhedral structures in mathematics and engineering.

4. Using the Calculator

Tips: Enter the number of vertices (must be ≥3) and the edge length in meters. All values must be valid positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is the minimum number of vertices for an antiprism?
A: The minimum number of vertices for an antiprism is 3, which forms a triangular antiprism.

Q2: How does the volume change with increasing vertices?
A: As the number of vertices increases, the antiprism becomes more cylindrical and the volume increases proportionally with the cube of the edge length.

Q3: Can this formula be used for all types of antiprisms?
A: Yes, this general formula applies to all regular antiprisms with any number of vertices.

Q4: What are the units of measurement?
A: The edge length should be entered in meters, and the volume result will be in cubic meters (m³).

Q5: Are there any limitations to this formula?
A: This formula assumes a regular antiprism with equal edge lengths and perfect geometric symmetry.

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