Height Of Hexadecagon Given Circumradius Calculator

Height of Hexadecagon Formula:

\[ h = \frac{\sin\left(\frac{7\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{r_c}{\frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2}} \]

Height of Hexadecagon (h):

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1. What is the Height of Hexadecagon Formula?

The Height of Hexadecagon formula calculates the vertical height of a regular hexadecagon (16-sided polygon) given its circumradius. This geometric relationship is derived from trigonometric principles and the properties of regular polygons.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \frac{\sin\left(\frac{7\pi}{16}\right)}{\sin\left(\frac{\pi}{16}\right)} \times \frac{r_c}{\frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2}} \]

Where:

\( h \) — Height of Hexadecagon (m)
\( r_c \) — Circumradius of Hexadecagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \sqrt{} \) — Square root function

Explanation: The formula combines trigonometric ratios with geometric properties of the hexadecagon to determine its height from the given circumradius.

3. Importance of Height Calculation

Details: Calculating the height of a hexadecagon is essential in geometric design, architectural planning, and engineering applications where precise dimensions of regular polygons are required.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and valid. The calculator will compute the corresponding height of the hexadecagon.

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 circumradius?
A: Circumradius is the radius of a circle that passes through all the vertices of a polygon. For regular polygons, this is the distance from the center to any vertex.

Q3: Why are trigonometric functions used in this formula?
A: Trigonometric functions help relate the angles and sides of the triangles formed within the hexadecagon, allowing calculation of the height from the circumradius.

Q4: Can this formula be used for irregular hexadecagons?
A: No, this formula is specifically designed for 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 requiring precise geometric measurements of 16-sided figures.