Height Of Rotunda Given Total Surface Area Calculator

Height of Rotunda Formula:

\[ h = \sqrt{1+\frac{2}{\sqrt{5}}} \times \sqrt{\frac{TSA}{\frac{1}{2} \times \left(5\sqrt{3} + \sqrt{10(65+29\sqrt{5})}\right)}} \]

Height of Rotunda (h):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Height of Rotunda Formula?

The Height of Rotunda formula calculates the vertical height of a Johnson solid known as a rotunda, given its total surface area. This geometric formula provides an accurate measurement based on the surface area of this specific polyhedron.

2. How Does the Calculator Work?

The calculator uses the Height of Rotunda formula:

\[ h = \sqrt{1+\frac{2}{\sqrt{5}}} \times \sqrt{\frac{TSA}{\frac{1}{2} \times \left(5\sqrt{3} + \sqrt{10(65+29\sqrt{5})}\right)}} \]

Where:

\( h \) — Height of Rotunda (m)
\( TSA \) — Total Surface Area of Rotunda (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)
\( \sqrt{3} \) — Square root of 3 (approximately 1.73205)

Explanation: The formula derives from the geometric properties of the rotunda, a Johnson solid with pentagonal and decagonal faces.

3. Importance of Height Calculation

Details: Calculating the height of a rotunda is essential in architectural design, geometric modeling, and mathematical applications involving Johnson solids. It helps determine the vertical dimension when the surface area is known.

4. Using the Calculator

Tips: Enter the total surface area of the rotunda in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a rotunda in geometry?
A: A rotunda is a Johnson solid (J6) consisting of a pentagon, decagon, and triangles, forming a cupola-like structure with specific geometric properties.

Q2: Why does the formula contain square roots?
A: The square roots come from the mathematical relationships between the edges, faces, and angles in the rotunda's geometric structure.

Q3: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Johnson solid known as a rotunda. Other polyhedra have different height formulas.

Q4: What are practical applications of this calculation?
A: This calculation is used in architectural design, 3D modeling, mathematical education, and geometric research involving Johnson solids.

Q5: How accurate is this formula?
A: The formula is mathematically exact for a perfect rotunda shape, providing precise height calculations when the surface area is known.