Edge Length of Antiprism given Volume Calculator

Formula Used:

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

Edge Length:

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1. What is the Edge Length of Antiprism?

The edge length of an antiprism is defined as the straight line joining the adjacent vertices of the antiprism. It is a fundamental geometric property that determines the size and proportions of the antiprism structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

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

Where:

\( N \) — Number of vertices of the antiprism
\( V \) — Volume of the antiprism (m³)
\( \pi \) — Mathematical constant pi (≈3.14159)
\( \sin \) — Sine trigonometric function
\( \cos \) — Cosine trigonometric function
\( \sqrt \) — Square root function

Explanation: This formula calculates the edge length of a regular antiprism based on its number of vertices and volume, using trigonometric relationships inherent in the antiprism's geometric structure.

3. Importance of Edge Length Calculation

Details: Calculating the edge length is essential for understanding the geometric properties of antiprisms, which are important in various fields including crystallography, molecular modeling, and architectural design. The edge length determines the overall dimensions and spatial relationships within the antiprism structure.

4. Using the Calculator

Tips: Enter the number of vertices (must be at least 3) and the volume (must be positive). The calculator will compute the corresponding edge length using the mathematical formula above.

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: Why is the number of vertices important?
A: The number of vertices determines the specific geometric properties and trigonometric relationships within the antiprism structure, affecting the edge length calculation.

Q3: What are typical applications of antiprisms?
A: Antiprisms are used in various fields including chemistry (molecular structures), architecture (geometric designs), and mathematics (polyhedral studies).

Q4: Are there limitations to this formula?
A: This formula applies specifically to regular antiprisms. For irregular antiprisms or other polyhedral structures, different formulas would be required.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for regular antiprisms, assuming precise input values and proper implementation of the trigonometric functions.