Volume of Tetrahedron Given Height Calculator

Volume of Tetrahedron Formula:

\[ V = \frac{(\sqrt{\frac{3}{2}} \times h)^3}{6 \times \sqrt{2}} \]

Volume of Tetrahedron:

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1. What is the Volume of Tetrahedron Given Height Formula?

The formula calculates the volume of a regular tetrahedron when its height is known. A tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{(\sqrt{\frac{3}{2}} \times h)^3}{6 \times \sqrt{2}} \]

Where:

\( V \) — Volume of the tetrahedron (m³)
\( h \) — Height of the tetrahedron (m)
\( \sqrt{} \) — Square root function

Explanation: The formula derives from the geometric properties of a regular tetrahedron, relating its volume to its height through mathematical constants and operations.

3. Importance of Volume Calculation

Details: Calculating the volume of a tetrahedron is essential in various fields including geometry, engineering, architecture, and 3D modeling. It helps in determining the space occupied by tetrahedral structures.

4. Using the Calculator

Tips: Enter the height of the tetrahedron in meters. The value must be positive and valid. The calculator will compute the volume using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular tetrahedron?
A: A regular tetrahedron is a tetrahedron where all four faces are equilateral triangles. All edges have the same length and all faces are congruent.

Q2: How is this formula derived?
A: The formula is derived from the relationship between the height and edge length of a regular tetrahedron, combined with the standard volume formula for pyramids.

Q3: What are the units for the result?
A: The volume is calculated in cubic meters (m³). If you input height in different units, make sure to convert the result accordingly.

Q4: Can this calculator handle irregular tetrahedrons?
A: No, this calculator is specifically designed for regular tetrahedrons where all edges are equal and all faces are congruent equilateral triangles.

Q5: What is the relationship between height and edge length?
A: In a regular tetrahedron, the height \( h \) is related to the edge length \( a \) by the formula: \( h = a \times \sqrt{\frac{2}{3}} \).