Diagonal of Hexadecagon across Five Sides given Diagonal across Seven Sides Calculator

Formula Used:

\[ Diagonal across Five Sides of Hexadecagon = Diagonal across Seven Sides of Hexadecagon \times \frac{\sin\left(\frac{5\pi}{16}\right)}{\sin\left(\frac{7\pi}{16}\right)} \]

Diagonal across Five Sides of Hexadecagon:

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1. What is Diagonal across Five Sides of Hexadecagon?

Diagonal across Five Sides of Hexadecagon is the straight line joining two non-adjacent vertices across five sides of a regular hexadecagon (16-sided polygon).

2. How Does the Calculator Work?

The calculator uses the trigonometric formula:

\[ d5 = d7 \times \frac{\sin\left(\frac{5\pi}{16}\right)}{\sin\left(\frac{7\pi}{16}\right)} \]

Where:

\( d5 \) — Diagonal across Five Sides of Hexadecagon (m)
\( d7 \) — Diagonal across Seven Sides of Hexadecagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function

3. Formula Explanation

Details: This formula utilizes the trigonometric properties of a regular hexadecagon. The ratio of sines of specific angles (5π/16 and 7π/16) relates the lengths of diagonals spanning different numbers of sides in the polygon.

4. Using the Calculator

Tips: Enter the length of the diagonal across seven sides in meters. The value must be positive and greater than zero. The calculator will compute the corresponding diagonal across five sides.

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: Why use trigonometric functions for this calculation?
A: Trigonometric functions help establish relationships between different diagonals in regular polygons through their central angles.

Q3: Can this formula be used for irregular hexadecagons?
A: No, this formula applies only to 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, architectural design, and engineering where regular polygonal shapes are employed.

Q5: How accurate is the calculation?
A: The calculation is mathematically exact for a perfect regular hexadecagon, using precise trigonometric values.