Total Surface Area of Dodecahedron given Circumsphere Radius Calculator

Formula Used:

\[ TSA = 3 \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^2 \]

Total Surface Area of Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Total Surface Area of Dodecahedron?

The Total Surface Area of a Dodecahedron is the total quantity of plane enclosed by the entire surface of this polyhedron. A dodecahedron is a three-dimensional shape with 12 regular pentagonal faces, 20 vertices, and 30 edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = 3 \times \sqrt{25 + (10 \times \sqrt{5})} \times \left( \frac{4 \times r_c}{\sqrt{3} \times (1 + \sqrt{5})} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Dodecahedron (m²)
\( r_c \) — Circumsphere Radius of Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: This formula calculates the total surface area of a regular dodecahedron based on the radius of its circumscribed sphere.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific applications. It helps in material estimation, heat transfer calculations, and structural analysis.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the total surface area of the dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a polyhedron with 12 flat faces, each being a regular pentagon. It is one of the five Platonic solids.

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that contains the dodecahedron such that all vertices lie on the sphere's surface.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula is specifically for regular dodecahedrons where all faces are congruent regular pentagons.

Q4: What are the practical applications of dodecahedrons?
A: Dodecahedrons are used in various fields including geometry education, dice design, architectural structures, and molecular modeling.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons. The accuracy depends on the precision of the input values.