Total Surface Area of Tetrahedron given Insphere Radius Calculator

Formula Used:

\[ TSA = \sqrt{3} \times (2 \times \sqrt{6} \times r_i)^2 \]

Total Surface Area of Tetrahedron (TSA):

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1. What is Total Surface Area of Tetrahedron?

The Total Surface Area of a Tetrahedron is the total quantity of plane enclosed by the entire surface of the Tetrahedron. It represents the sum of the areas of all four triangular faces of the tetrahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \sqrt{3} \times (2 \times \sqrt{6} \times r_i)^2 \]

Where:

\( TSA \) — Total Surface Area of Tetrahedron (m²)
\( r_i \) — Insphere Radius of Tetrahedron (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{6} \) — Square root of 6 (approximately 2.449)

Explanation: The formula calculates the total surface area of a regular tetrahedron based on the radius of its inscribed sphere (insphere).

3. Importance of Surface Area Calculation

Details: Calculating the surface area of a tetrahedron is important in various geometric applications, 3D modeling, material science calculations, and engineering design where tetrahedral structures are used.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular tetrahedron?
A: A regular tetrahedron is a polyhedron with four equilateral triangular faces, four vertices, and six edges where all edges are equal in length.

Q2: What is the insphere radius of a tetrahedron?
A: The insphere radius is the radius of the sphere that is contained by the tetrahedron in such a way that all the faces just touch the sphere.

Q3: Can this formula be used for irregular tetrahedrons?
A: No, this specific formula applies only to regular tetrahedrons where all faces are equilateral triangles and all edges are equal.

Q4: What are the units for surface area?
A: The surface area is typically measured in square meters (m²), but any square unit can be used as long as consistency is maintained with the radius units.

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