Area of Hexadecagon given Inradius Calculator

Formula Used:

\[ A = 4 \times \left( \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2 \times (2 + \sqrt{2})}} \right)^2 \times \cot\left(\frac{\pi}{16}\right) \]

Area of Hexadecagon:

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1. What is the Area of Hexadecagon given Inradius Formula?

The formula calculates the area of a regular hexadecagon (16-sided polygon) when the inradius (radius of the inscribed circle) is known. This geometric formula provides an accurate measurement of the 2-dimensional space occupied by the hexadecagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = 4 \times \left( \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2 \times (2 + \sqrt{2})}} \right)^2 \times \cot\left(\frac{\pi}{16}\right) \]

Where:

\( A \) — Area of Hexadecagon (m²)
\( r_i \) — Inradius of Hexadecagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sqrt{} \) — Square root function
\( \cot \) — Cotangent trigonometric function

Explanation: The formula derives from the geometric properties of regular polygons, specifically relating the inradius to the side length and ultimately to the area through trigonometric relationships.

3. Importance of Area Calculation

Details: Calculating the area of geometric shapes is fundamental in mathematics, engineering, architecture, and various scientific fields. For regular polygons like hexadecagons, area calculations help in material estimation, structural design, and spatial planning.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and greater than zero. The calculator will compute the area using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the inradius of a polygon?
A: The inradius is the radius of the largest circle that can fit inside the polygon, tangent to all its sides.

Q3: Why use this specific formula?
A: This formula provides a direct relationship between the inradius and area of a regular hexadecagon, making calculations efficient and accurate.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters first, then convert the result back to your desired unit.

Q5: How accurate is the calculation?
A: The calculation uses precise mathematical constants and functions, providing results accurate to six decimal places.