Diagonal of Hexadecagon across Three Sides given Circumradius Calculator

Formula Used:

\[ Diagonal_{across\ 3\ sides} = \frac{\sin\left(\frac{3\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{Circumradius}{\frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2}} \]

Diagonal across Three Sides:

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1. What is Diagonal across Three Sides of Hexadecagon?

The diagonal across three sides of a hexadecagon is the straight line joining two non-adjacent vertices that are separated by three sides of the regular 16-sided polygon. It represents one of the longer diagonals in the hexadecagon geometry.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ Diagonal_{across\ 3\ sides} = \frac{\sin\left(\frac{3\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{Circumradius}{\frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2}} \]

Where:

\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \sqrt \) — Square root function
\( Circumradius \) — Radius of the circumscribed circle

3. Mathematical Formula Explanation

Details: The formula combines trigonometric functions (sine) with geometric relationships specific to the regular hexadecagon. The complex square root expression represents the geometric relationship between the circumradius and the diagonal length across three sides of the polygon.

4. Using the Calculator

Tips: Enter the circumradius of the hexadecagon in meters. The circumradius must be a positive value greater than zero. The calculator will compute the diagonal length across three sides of the regular hexadecagon.

5. Frequently Asked Questions (FAQ)

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

Q2: What is circumradius?
A: Circumradius is the radius of a circle that passes through all vertices of a polygon. For regular polygons, this circle is called the circumcircle.

Q3: How many diagonals does a hexadecagon have?
A: A hexadecagon has 104 diagonals in total, with different lengths depending on how many sides they cross.

Q4: What are the practical applications of this calculation?
A: This calculation is used in geometry, architecture, engineering design, and computer graphics where regular polygonal shapes are employed.

Q5: 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.