Inradius of Hexadecagon given Diagonal across Five Sides Calculator

Formula Used:

\[ Inradius = \frac{Diagonal \times \sin(\pi/16)}{\sin(5\pi/16)} \times \frac{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}}{2} \]

Inradius of Hexadecagon:

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1. What is Inradius of Hexadecagon?

The inradius of a hexadecagon is defined as the radius of the circle which is inscribed inside the hexadecagon, touching all its sides. A hexadecagon is a polygon with 16 sides and 16 angles.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Inradius = \frac{Diagonal \times \sin(\pi/16)}{\sin(5\pi/16)} \times \frac{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}}{2} \]

Where:

\( Diagonal \) — Diagonal across five sides of hexadecagon (meters)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \sqrt \) — Square root function

3. Formula Explanation

Details: This formula calculates the inradius of a regular hexadecagon based on the diagonal that spans five sides. It combines trigonometric functions with geometric constants specific to the 16-sided polygon.

4. Using the Calculator

Tips: Enter the diagonal across five sides of the hexadecagon in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a hexadecagon?
A: A hexadecagon is a polygon with 16 sides and 16 angles. It is also known as a 16-gon.

Q2: What does the inradius represent?
A: The inradius represents the radius of the largest circle that can fit inside the hexadecagon, touching all its sides.

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

Q4: What are the practical applications of this calculation?
A: This calculation is useful in geometry, architecture, engineering design, and any field dealing with regular polygonal shapes.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular hexadecagons, using exact trigonometric relationships and constants.