Area of Decagon Given Diagonal Across Three Sides Calculator

Area of Decagon Formula:

\[ A = \frac{5}{2} \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{2 \times d3}{\sqrt{14 + (6 \times \sqrt{5})}} \right)^2 \]

Area of Decagon:

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1. What is the Area of Decagon Formula?

The formula calculates the area of a regular decagon (10-sided polygon) when given the diagonal length across three sides. This formula provides an accurate measurement of the 2-dimensional space occupied by the decagon.

2. How Does the Calculator Work?

The calculator uses the decagon area formula:

\[ A = \frac{5}{2} \times \sqrt{5 + (2 \times \sqrt{5})} \times \left( \frac{2 \times d3}{\sqrt{14 + (6 \times \sqrt{5})}} \right)^2 \]

Where:

\( A \) — Area of the decagon
\( d3 \) — Diagonal across three sides of the decagon
\( \sqrt{} \) — Square root function

Explanation: The formula accounts for the geometric properties of a regular decagon and uses the diagonal measurement to compute the total area.

3. Importance of Area Calculation

Details: Calculating the area of geometric shapes is fundamental in mathematics, engineering, architecture, and various design fields. Accurate area calculations are essential for material estimation, space planning, and structural design.

4. Using the Calculator

Tips: Enter the diagonal across three sides measurement 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 10-sided polygon where all sides are equal in length and all interior angles are equal (144 degrees each).

Q2: How is the diagonal across three sides measured?
A: The diagonal across three sides connects two vertices that are separated by two vertices between them in a regular decagon.

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

Q4: What units should I use for the diagonal measurement?
A: The calculator accepts any consistent unit (meters, centimeters, inches, etc.), but the result will be in square units of whatever unit you input.

Q5: Are there other ways to calculate decagon area?
A: Yes, decagon area can also be calculated using side length, apothem, or other diagonal measurements, but this formula specifically uses the diagonal across three sides.