Total Surface Area Of Small Stellated Dodecahedron Given Circumradius Calculator

Formula Used:

\[ TSA = 15 \times \sqrt{5 + 2\sqrt{5}} \times \left( \frac{4 \times r_c}{\sqrt{50 + 22\sqrt{5}}} \right)^2 \]

Total Surface Area of Small Stellated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

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, providing a measure of the polyhedron's external coverage.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 15 \times \sqrt{5 + 2\sqrt{5}} \times \left( \frac{4 \times r_c}{\sqrt{50 + 22\sqrt{5}}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Small Stellated Dodecahedron (m²)
\( r_c \) — Circumradius of Small Stellated Dodecahedron (m)
\( \sqrt{5 + 2\sqrt{5}} \) — Mathematical constant related to pentagon geometry
\( \sqrt{50 + 22\sqrt{5}} \) — Mathematical constant derived from dodecahedron properties

Explanation: The formula calculates the total surface area based on the circumradius, incorporating geometric constants specific to the Small Stellated Dodecahedron's structure.

3. Importance of Total Surface Area Calculation

Details: Calculating the total surface area is essential for understanding the polyhedron's geometric properties, material requirements for construction, and various applications in mathematics, architecture, and design.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and valid. The calculator will compute the total surface area based on the provided circumradius.

5. Frequently Asked Questions (FAQ)

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

Q2: How is circumradius related to surface area?
A: The circumradius (distance from center to any vertex) determines the size of the polyhedron, which directly affects its surface area through the mathematical relationship shown in the formula.

Q3: What are typical values for circumradius?
A: The circumradius can vary widely depending on the specific application, but it's typically a positive real number measured in meters or appropriate length units.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Small Stellated Dodecahedron. Other polyhedra have different surface area formulas based on their unique geometric properties.

Q5: What precision should I expect from the calculation?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most mathematical and engineering applications involving geometric calculations.