Total Surface Area of Small Stellated Dodecahedron Calculator

Total Surface Area Formula:

\[ TSA = 15 \times \sqrt{5 + 2\sqrt{5}} \times (Edge\ Length)^2 \]

Total Surface Area:

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1. What is the Total Surface Area of Small Stellated Dodecahedron?

The Small Stellated Dodecahedron is a Kepler-Poinsot polyhedron with 12 pentagram faces. Its total surface area represents the sum of the areas of all its faces, which is an important geometric property for various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 + 2\sqrt{5}} \times (Edge\ Length)^2 \]

Where:

\( TSA \) — Total Surface Area
\( Edge\ Length \) — Length of one edge of the polyhedron
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the total surface area by multiplying a constant derived from the geometry of pentagrams (15 × √(5 + 2√5)) by the square of the edge length.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is crucial in various fields including architecture, materials science, and 3D modeling. For the Small Stellated Dodecahedron, this measurement helps in understanding its geometric properties and applications in mathematical art and design.

4. Using the Calculator

Tips: Enter the edge length in meters. The value must be positive and greater than zero. The calculator will compute the total surface area in square meters.

5. Frequently Asked Questions (FAQ)

Q1: What is a Small Stellated Dodecahedron?
A: It's one of the four Kepler-Poinsot solids, formed by extending the faces of a regular dodecahedron until they intersect, creating a star-shaped polyhedron with 12 pentagram faces.

Q2: What are the units for the result?
A: The result is in square meters (m²), which is consistent with the input edge length unit (meters).

Q3: Can this calculator handle different units?
A: The calculator uses consistent units. If you input edge length in centimeters, the result will be in square centimeters.

Q4: What is the precision of the calculation?
A: The calculator provides results with 6 decimal places precision, suitable for most mathematical and engineering applications.

Q5: Are there any limitations to this calculation?
A: The formula assumes a perfect geometric shape with all edges equal and faces perfectly formed as regular pentagrams.