Inradius of Hexadecagon given Area Calculator

Formula Used:

\[ Inradius\ of\ Hexadecagon = \frac{1+\sqrt{2}+\sqrt{2(2+\sqrt{2})}}{2} \times \sqrt{\frac{Area\ of\ Hexadecagon}{4 \times \cot(\frac{\pi}{16})}} \]

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 to any side of the hexadecagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Inradius = \frac{1+\sqrt{2}+\sqrt{2(2+\sqrt{2})}}{2} \times \sqrt{\frac{Area}{4 \times \cot(\frac{\pi}{16})}} \]

Where:

\( Area \) — Area of the hexadecagon (m²)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sqrt{2} \) — Square root of 2 (approximately 1.41421)
\( \cot(\frac{\pi}{16}) \) — Cotangent of π/16

3. Formula Explanation

Details: This formula derives from the geometric properties of a regular hexadecagon. The constant factor accounts for the specific shape characteristics, while the square root term relates the area to the inradius through trigonometric functions.

4. Using the Calculator

Tips: Enter the area of the hexadecagon in square meters. The area must be a positive value. The calculator will compute the inradius based on the mathematical relationship between area and inradius for a 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: Why is the formula so complex?
A: The complexity arises from the trigonometric relationships in a 16-sided polygon. The formula incorporates the specific geometric properties of a regular hexadecagon.

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 this calculation?
A: This calculation is useful in geometry, architecture, engineering, and design where hexadecagonal shapes are used.

Q5: How accurate is the calculation?
A: The calculation is mathematically precise for regular hexadecagons, with accuracy limited only by floating-point precision.