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Diagonal of Decagon across Three Sides given Inradius Calculator

Formula Used:

\[ d_3 = \frac{\sqrt{14 + 6\sqrt{5}}}{2} \times \frac{2 \times r_i}{\sqrt{5 + 2\sqrt{5}}} \]

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1. What is Diagonal across Three Sides of Decagon?

The diagonal across three sides of a decagon is a straight line joining two non-adjacent vertices that spans three sides of the regular decagon. It represents one of the longer diagonals in a decagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_3 = \frac{\sqrt{14 + 6\sqrt{5}}}{2} \times \frac{2 \times r_i}{\sqrt{5 + 2\sqrt{5}}} \]

Where:

3. Formula Explanation

Details: This formula calculates the length of the diagonal that spans three sides of a regular decagon based on its inradius (the radius of the inscribed circle). The formula incorporates mathematical constants derived from the geometric properties of a regular decagon.

4. Using the Calculator

Tips: Enter the inradius of the decagon in meters. The inradius must be a positive value. The calculator will compute the diagonal length across three sides of the decagon.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular decagon?
A: A regular decagon is a polygon with 10 equal sides and 10 equal angles.

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

Q3: What are the practical applications of this calculation?
A: This calculation is useful in geometry, architecture, engineering, and design where decagonal shapes are used.

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: How accurate is this calculation?
A: The calculation is mathematically exact for regular decagons, though practical measurements may have slight variations.

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