Inradius of Decagon Given Circumradius Calculator

Inradius of Decagon Formula:

\[ r_i = \frac{\sqrt{5 + 2\sqrt{5}}}{2} \times \frac{2r_c}{1 + \sqrt{5}} \]

Inradius of Decagon (r_i):

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1. What is Inradius of Decagon?

The inradius of a decagon is the radius of the inscribed circle that touches all ten sides of the regular decagon from the inside. It represents the distance from the center of the decagon to any of its sides.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{5 + 2\sqrt{5}}}{2} \times \frac{2r_c}{1 + \sqrt{5}} \]

Where:

\( r_i \) — Inradius of Decagon
\( r_c \) — Circumradius of Decagon
\( \sqrt{5 + 2\sqrt{5}} \) — Mathematical constant for decagon geometry
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the inradius of a regular decagon based on its circumradius, using geometric properties and mathematical constants specific to decagonal shapes.

3. Importance of Inradius Calculation

Details: Calculating the inradius is essential in geometry for determining the size of the inscribed circle, which is crucial for various applications in mathematics, engineering, and design involving decagonal shapes.

4. Using the Calculator

Tips: Enter the circumradius of the decagon in meters. The value must be positive and greater than zero. The calculator will compute the corresponding inradius.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular decagon?
A: A regular decagon is a ten-sided polygon with all sides equal in length and all interior angles equal (144 degrees each).

Q2: How is inradius different from circumradius?
A: Inradius is the radius of the inscribed circle (touching the sides), while circumradius is the radius of the circumscribed circle (passing through the vertices).

Q3: What are practical applications of this calculation?
A: This calculation is used in architectural design, engineering projects, and mathematical modeling where decagonal shapes are involved.

Q4: Can this formula be used for irregular decagons?
A: No, this formula applies only to regular decagons where all sides and angles are equal.

Q5: What is the relationship between inradius and side length?
A: For a regular decagon, the inradius can also be calculated from the side length using a different formula: \( r_i = \frac{s}{2} \times \sqrt{5 + 2\sqrt{5}} \).