Inradius of Nonagon given Circumradius Calculator

Formula Used:

\[ Inradius\ of\ Nonagon = Circumradius\ of\ Nonagon \times \frac{\sin(\pi/9)}{\tan(\pi/9)} \]

Inradius of Nonagon:

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1. What is the Inradius of Nonagon given Circumradius?

The inradius of a nonagon is the radius of the inscribed circle that touches all sides of the nonagon. This calculator computes the inradius when the circumradius (radius of the circumscribed circle) is known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Inradius\ of\ Nonagon = Circumradius\ of\ Nonagon \times \frac{\sin(\pi/9)}{\tan(\pi/9)} \]

Where:

\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \tan \) — Tangent trigonometric function
Circumradius of Nonagon — Radius of the circumscribed circle

Explanation: The formula derives from the geometric properties of a regular nonagon, using trigonometric relationships between the circumradius and inradius.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry for determining the size of the largest circle that fits inside a nonagon, which has applications in design, architecture, and various engineering fields.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumradius and inradius in a regular nonagon?
A: In a regular nonagon, the inradius is always smaller than the circumradius, and their ratio is constant for all regular nonagons.

Q2: Can this formula be used for irregular nonagons?
A: No, this formula applies only to regular nonagons where all sides and angles are equal.

Q3: What are practical applications of knowing the inradius?
A: The inradius helps in determining the maximum size of objects that can fit inside a nonagonal shape, useful in manufacturing and design.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for regular nonagons, limited only by the precision of the input values and computational floating-point arithmetic.

Q5: Can I calculate circumradius if I know the inradius?
A: Yes, the formula can be rearranged to solve for circumradius given the inradius: \( Circumradius = Inradius \times \frac{\tan(\pi/9)}{\sin(\pi/9)} \)