Circumradius of Nonagon given Diagonal across Three Sides Calculator

Formula Used:

\[ r_c = \frac{d_3}{2 \cdot \sin\left(\frac{3\pi}{9}\right)} \]

Circumradius of Nonagon (rₑ):

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1. What is the Circumradius of Nonagon?

The Circumradius of Nonagon is the radius of a circumcircle touching each of the vertices of the Nonagon. It is a fundamental geometric property that helps in understanding the spatial dimensions of a regular nonagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{d_3}{2 \cdot \sin\left(\frac{3\pi}{9}\right)} \]

Where:

\( r_c \) — Circumradius of Nonagon (m)
\( d_3 \) — Diagonal across Three Sides of Nonagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function

Explanation: This formula calculates the circumradius based on the diagonal that spans three sides of a regular nonagon, utilizing trigonometric relationships inherent in the polygon's geometry.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential in geometry for determining the size of the circumscribed circle around a nonagon, which has applications in various fields including architecture, engineering, and design where precise geometric relationships are required.

4. Using the Calculator

Tips: Enter the diagonal across three sides of the nonagon in meters. The value must be positive and greater than zero for accurate calculation.

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 is the diagonal across three sides defined?
A: The diagonal across three sides is the straight line joining two non-adjacent vertices which spans across three sides of the nonagon.

Q3: Can this calculator be used for irregular nonagons?
A: No, this formula and calculator are specifically designed for regular nonagons where all sides and angles are equal.

Q4: What are practical applications of circumradius calculation?
A: Circumradius calculations are used in construction, mechanical design, computer graphics, and any field requiring precise geometric modeling of polygonal shapes.

Q5: How accurate is the trigonometric calculation?
A: The calculation uses PHP's built-in trigonometric functions which provide high precision, typically accurate to many decimal places for practical applications.