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Circumradius of Hexadecagon Calculator

Circumradius of Hexadecagon Formula:

\[ r_c = \frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2} \times S \]

m

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1. What is the Circumradius of Hexadecagon?

The circumradius of a hexadecagon (16-sided polygon) is the radius of a circumcircle that touches all vertices of the hexadecagon. It's an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the circumradius formula:

\[ r_c = \frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2} \times S \]

Where:

Explanation: The formula calculates the circumradius based on the side length of a regular hexadecagon, incorporating nested square roots to achieve the precise geometric relationship.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential in geometry, architecture, and engineering for designing and analyzing 16-sided structures, creating precise geometric constructions, and solving related mathematical problems.

4. Using the Calculator

Tips: Enter the side length of the hexadecagon in meters. The value must be positive and greater than zero. The calculator will compute the circumradius using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a hexadecagon?
A: A hexadecagon is a polygon with 16 sides and 16 angles. When all sides and angles are equal, it's called a regular hexadecagon.

Q2: How is the circumradius different from the inradius?
A: The circumradius is the radius of the circle that passes through all vertices, while the inradius is the radius of the circle inscribed within the polygon that touches all sides.

Q3: Can this formula be used for irregular hexadecagons?
A: No, this formula is specifically for regular hexadecagons where all sides and angles are equal. Irregular hexadecagons require different calculation methods.

Q4: What are practical applications of hexadecagons?
A: Hexadecagons are used in architecture, design, coin minting (some coins have 16 sides), and various engineering applications where specific geometric properties are needed.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise, using the exact formula for a regular hexadecagon. The result is rounded to 10 decimal places for practical use.

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