Edge Length of Antiprism given Surface to Volume Ratio Calculator

Formula Used:

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

Edge Length of Antiprism:

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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:

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

Where:

\( l_e \) — Edge Length of Antiprism (m)
\( N_{Vertices} \) — Number of Vertices of Antiprism
\( R_{A/V} \) — Surface to Volume Ratio of Antiprism (m⁻¹)
\( \pi \) — Archimedes' constant (approximately 3.14159)

3. Formula Explanation

Details: This formula calculates the edge length of an antiprism based on the number of vertices and the surface-to-volume ratio. It incorporates trigonometric functions (sine, cosine, cotangent) and square roots to accurately determine the geometric relationship between these parameters.

4. Using the Calculator

Tips: Enter the number of vertices (must be ≥3) and the surface-to-volume ratio (must be >0). The calculator will compute the corresponding edge length of the antiprism in meters.

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 is the number of vertices important?
A: The number of vertices determines the specific geometry and symmetry properties of the antiprism, affecting all its dimensional relationships.

Q3: What does surface-to-volume ratio represent?
A: The surface-to-volume ratio indicates how much surface area the antiprism has relative to its volume, which is important for various physical and engineering applications.

Q4: Are there limitations to this formula?
A: The formula assumes a regular antiprism and may not apply to irregular or modified antiprism structures.

Q5: What units are used in the calculation?
A: The edge length is returned in meters, and surface-to-volume ratio should be provided in reciprocal meters (m⁻¹).