Height of Tetragonal Trapezohedron given Short Edge Calculator

Formula Used:

\[ h = \sqrt{\frac{1}{2}(4+3\sqrt{2})} \times \frac{l_{short}}{\sqrt{\sqrt{2}-1}} \]

Height of Tetragonal Trapezohedron:

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1. What is the Height of Tetragonal Trapezohedron?

The height of a tetragonal trapezohedron is the distance between the two peak vertices where the long edges join. It's a crucial measurement in understanding the three-dimensional geometry of this polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \sqrt{\frac{1}{2}(4+3\sqrt{2})} \times \frac{l_{short}}{\sqrt{\sqrt{2}-1}} \]

Where:

\( h \) — Height of Tetragonal Trapezohedron (m)
\( l_{short} \) — Short Edge of Tetragonal Trapezohedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the height based on the length of the short edge, using the mathematical relationship derived from the geometry of the tetragonal trapezohedron.

3. Importance of Height Calculation

Details: Calculating the height is essential for understanding the complete dimensions of a tetragonal trapezohedron, which is important in crystallography, materials science, and geometric modeling applications.

4. Using the Calculator

Tips: Enter the length of the short edge in meters. The value must be positive and greater than zero. The calculator will compute the corresponding height of the tetragonal trapezohedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a tetragonal trapezohedron?
A: A tetragonal trapezohedron is a polyhedron with eight faces that are congruent kites, forming a shape that resembles two square pyramids base-to-base but rotated 45 degrees relative to each other.

Q2: What are the practical applications of this calculation?
A: This calculation is useful in crystallography for determining crystal structures, in materials science for nanoparticle characterization, and in 3D modeling for accurate geometric representations.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to tetragonal trapezohedra. Other polyhedra have different geometric relationships between their edges and height.

Q4: What precision should I use for the input values?
A: For most applications, 4-6 decimal places of precision are sufficient. The calculator accepts up to 4 decimal places by default but can handle more precise inputs.

Q5: How does the height relate to other dimensions of the tetragonal trapezohedron?
A: The height is directly proportional to the short edge length, with the proportionality constant derived from the specific geometry of the tetragonal trapezohedron.