Inradius of Decagon given Diagonal across Three Sides Calculator

Inradius of Decagon Formula:

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

Inradius of Decagon (r_i):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Inradius of Decagon given Diagonal across Three Sides?

The inradius of a decagon is the radius of the incircle that fits perfectly inside the decagon, touching all its sides. This calculator determines the inradius when the diagonal across three sides of the decagon is known.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_i \) — Inradius of Decagon (m)
\( d_3 \) — Diagonal across Three Sides of Decagon (m)

Explanation: The formula derives from geometric relationships in a regular decagon, using the diagonal measurement to calculate the radius of the inscribed circle.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry for determining properties of the incircle, area calculations, and various architectural and design applications involving decagonal shapes.

4. Using the Calculator

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

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 the diagonal across three sides defined?
A: The diagonal across three sides connects two non-adjacent vertices with exactly two vertices between them along the perimeter.

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

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 the calculation?
A: The calculation provides precise results based on mathematical relationships in regular decagons, with accuracy depending on the input measurement precision.