Circumradius of Hexadecagon given Diagonal across Three Sides Calculator

Formula Used:

\[ r_c = \frac{d_3 \cdot \sin(\pi/16)}{\sin(3\pi/16)} \cdot \frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2} \]

Circumradius of Hexadecagon:

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1. What is Circumradius of Hexadecagon?

The circumradius of a hexadecagon is the radius of a circumcircle that touches all sixteen vertices of the hexadecagon. It represents the distance from the center of the hexadecagon to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ r_c = \frac{d_3 \cdot \sin(\pi/16)}{\sin(3\pi/16)} \cdot \frac{\sqrt{4 + 2\sqrt{2} + \sqrt{20 + 14\sqrt{2}}}}{2} \]

Where:

\( r_c \) — Circumradius of Hexadecagon
\( d_3 \) — Diagonal across Three Sides of Hexadecagon
\( \pi \) — Archimedes' constant (approximately 3.14159)

3. Formula Explanation

Details: This formula derives from trigonometric relationships within a regular hexadecagon. The sine functions account for the angular relationships between vertices, while the square root expressions represent geometric properties specific to the 16-sided polygon.

4. Using the Calculator

Tips: Enter the diagonal measurement across three sides of the hexadecagon in meters. The value must be positive and non-zero. The calculator will compute the corresponding circumradius.

5. Frequently Asked Questions (FAQ)

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

Q2: How is the diagonal across three sides measured?
A: It's the straight line distance between two non-adjacent vertices that are separated by two vertices in between.

Q3: What are practical applications of this calculation?
A: This calculation is useful in geometry, architecture, engineering design, and any field dealing with regular polygonal structures.

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 units should be used for input?
A: The calculator accepts any consistent unit of length (meters, centimeters, inches, etc.), but the result will be in the same units as the input.