Inradius Of Hexadecagon Given Diagonal Across Four Sides Calculator

Formula Used:

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

Inradius of Hexadecagon:

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1. What is the Inradius of Hexadecagon?

The inradius of a hexadecagon (16-sided polygon) is the radius of the circle that fits perfectly inside the hexadecagon, touching all its sides. It represents the distance from the center of the hexadecagon to any of its sides.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( Diagonal \) — Diagonal across four sides of the hexadecagon (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sin(\pi/16) \) — Sine of π/16 radians
\( \pi \) — Mathematical constant pi (approximately 3.1416)

Explanation: This formula calculates the inradius based on the diagonal measurement across four sides, using trigonometric functions and square roots to account for the geometric properties of a regular hexadecagon.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry and engineering for determining the size of inscribed circles, designing components with specific clearance requirements, and understanding the spatial relationships within polygonal structures.

4. Using the Calculator

Tips: Enter the diagonal measurement across four sides of the hexadecagon in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: How is the diagonal across four sides measured?
A: The diagonal across four sides is the straight line distance between two non-adjacent vertices that are separated by three vertices between them.

Q3: Can this calculator 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 practical applications of inradius calculation?
A: Inradius calculations are used in mechanical engineering, architecture, and manufacturing for designing components with specific clearance requirements and optimal space utilization.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular hexadecagons, with accuracy depending on the precision of the input measurement.