1. What is the Circumradius of Decagon?

The circumradius of a decagon is the radius of a circumcircle that touches all ten vertices of the decagon. It's an important geometric property that relates to the size and proportions of the regular decagon.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumradius of the decagon
• \( h \) — Height of the decagon
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula relates the circumradius to the height of a regular decagon using the golden ratio and square root relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential for geometric constructions, architectural designs, and various engineering applications involving regular decagonal shapes.

4. Using the Calculator

Tips: Enter the height of the decagon in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular decagon?
A: A regular decagon is a ten-sided polygon with all sides equal and all interior angles equal (144° each).

Q2: How is the circumradius related to the side length?
A: For a regular decagon, the circumradius can also be expressed as \( r_c = \frac{s}{2} \times \sqrt{5 + 2\sqrt{5}} \), where s is the side length.

Q3: What is the significance of the golden ratio in this formula?
A: The term \( \frac{1 + \sqrt{5}}{2} \) represents the golden ratio (φ ≈ 1.618), which appears frequently in decagon geometry.

Q4: Can this formula be used for irregular decagons?
A: No, this formula applies only to regular decagons where all sides and angles are equal.

Q5: What are practical applications of decagon geometry?
A: Decagonal shapes are used in architecture, design, engineering, and various decorative patterns.