Side of Nonagon Given Height Calculator

Formula Used:

\[ S = \frac{2 \times \sin\left(\frac{\pi}{9}\right)}{1 + \cos\left(\frac{\pi}{9}\right)} \times h \]

Side of Nonagon:

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1. What is the Side of Nonagon Given Height Formula?

The formula calculates the side length of a regular nonagon (9-sided polygon) when the height (distance from center to a side) is known. It uses trigonometric relationships specific to the nonagon's geometry.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ S = \frac{2 \times \sin\left(\frac{\pi}{9}\right)}{1 + \cos\left(\frac{\pi}{9}\right)} \times h \]

Where:

\( S \) — Side length of the nonagon
\( h \) — Height of the nonagon
\( \pi \) — Mathematical constant pi (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \cos \) — Cosine trigonometric function

Explanation: The formula derives from the geometric properties of a regular nonagon, using trigonometric relationships between the central angle (40°) and the height measurement.

3. Importance of Side Calculation

Details: Calculating the side length from height is essential in geometry, architecture, and engineering applications involving nonagonal shapes. It helps in determining perimeter, area, and other geometric properties.

4. Using the Calculator

Tips: Enter the height of the nonagon in meters. The height must be a positive value greater than zero. The calculator will compute the corresponding side length.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular nonagon?
A: A regular nonagon is a nine-sided polygon where all sides are equal in length and all interior angles are equal (140° each).

Q2: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular nonagon, using precise trigonometric functions and the constant pi.

Q3: Can this formula be used for irregular nonagons?
A: No, this formula only applies to regular nonagons where all sides and angles are equal.

Q4: What are practical applications of this calculation?
A: This calculation is used in architectural design, mechanical engineering, and geometric pattern creation where nonagonal shapes are required.

Q5: How is the height measured in a nonagon?
A: The height is measured as the perpendicular distance from the center of the nonagon to the midpoint of any side.