Diagonal Of Hexadecagon Across Five Sides Given Inradius Calculator

Formula Used:

\[ d5 = \frac{\sin\left(\frac{5\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}} \]

Diagonal across Five Sides (d5):

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1. What is Diagonal across Five Sides of Hexadecagon?

The diagonal across five sides of a hexadecagon is the straight line joining two non-adjacent vertices that are separated by five sides of the regular 16-sided polygon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d5 = \frac{\sin\left(\frac{5\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}} \]

Where:

\( d5 \) — Diagonal across five sides
\( r_i \) — Inradius of the hexadecagon
\( \pi \) — Mathematical constant pi (≈3.14159)
\( \sin \) — Trigonometric sine function
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the diagonal length based on the geometric properties of a regular hexadecagon and its inscribed circle.

3. Importance of Diagonal Calculation

Details: Calculating diagonals in regular polygons is important for geometric analysis, architectural design, and various engineering applications where precise measurements are required.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and non-zero to get a valid calculation result.

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 the inradius of a polygon?
A: The inradius is the radius of the circle that can be inscribed inside the polygon, touching all its sides.

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 span.

Q4: What are the practical applications of this calculation?
A: This calculation is useful in geometry, architecture, mechanical design, and any field requiring precise polygonal measurements.

Q5: Can this formula be used for irregular hexadecagons?
A: No, this formula specifically applies to regular hexadecagons where all sides and angles are equal.