Volume of Antiprism Given Height Calculator

Volume of Antiprism Formula:

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

Volume of Antiprism:

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1. What is the Volume of Antiprism Formula?

The volume of an antiprism is calculated using a complex formula that takes into account the number of vertices and the height of the antiprism. This formula involves trigonometric functions to accurately determine the three-dimensional space enclosed by the antiprism structure.

2. How Does the Calculator Work?

The calculator uses the antiprism volume formula:

Where:

\( V \) — Volume of Antiprism (m³)
\( N \) — Number of Vertices of Antiprism
\( h \) — Height of Antiprism (m)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula combines trigonometric functions (sine, cosine, secant) and square roots to calculate the volume based on the geometric properties of the antiprism.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for understanding the capacity and spatial properties of antiprism structures, which have applications in geometry, architecture, and materials science.

4. Using the Calculator

Tips: Enter the number of vertices (minimum 3) and the height of the antiprism. All values must be valid positive numbers.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the minimum number of vertices required?
A: The minimum number of vertices for an antiprism is 3, which forms a triangular antiprism.

Q3: How accurate is this formula?
A: The formula provides exact mathematical calculation of antiprism volume based on geometric principles.

Q4: Can this calculator handle decimal inputs?
A: Yes, the calculator accepts decimal values for height, but vertices must be whole numbers.

Q5: What units should I use for height?
A: The height should be in meters, and the resulting volume will be in cubic meters (m³).