Diagonal Of Hexadecagon Across Three Sides Given Inradius Calculator

Formula Used:

\[ d3 = \frac{\sin\left(\frac{3\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}} \]

Diagonal across Three Sides of Hexadecagon:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Diagonal of Hexadecagon across Three Sides?

The diagonal across three sides of a hexadecagon is the straight line joining two non-adjacent vertices that span across three sides of the 16-sided polygon. It represents one of the several possible diagonals in a regular hexadecagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d3 = \frac{\sin\left(\frac{3\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{2 \times r_i}{1 + \sqrt{2} + \sqrt{2(2 + \sqrt{2})}} \]

Where:

\( d3 \) — Diagonal across three sides of hexadecagon
\( r_i \) — Inradius of the hexadecagon
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \sqrt{} \) — Square root function

Explanation: The formula derives from trigonometric relationships in a regular hexadecagon, using the inradius and specific angle ratios to calculate the diagonal length.

3. Importance of Diagonal Calculation

Details: Calculating diagonals in regular polygons is important in geometry, architecture, and engineering for determining spatial relationships, structural properties, and design parameters of polygonal shapes.

4. Using the Calculator

Tips: Enter the inradius value in meters. The inradius must be a positive number. The calculator will compute the diagonal length across three sides of the regular hexadecagon.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular hexadecagon?
A: A regular hexadecagon is a 16-sided polygon where all sides are equal in length and all interior angles are equal (157.5 degrees each).

Q2: 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 across.

Q3: What is the relationship between inradius and diagonal?
A: The inradius (radius of inscribed circle) is related to various diagonals through trigonometric functions based on the central angles of the polygon.

Q4: Can this formula be used for irregular hexadecagons?
A: No, this formula applies only to regular hexadecagons where all sides and angles are equal.

Q5: What are practical applications of this calculation?
A: This calculation is useful in architectural design, mechanical engineering, computer graphics, and any field dealing with regular polygonal structures.