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How to find the area of a decagon?

Understand the Problem

The question is asking how to calculate the area of a decagon, which is a ten-sided polygon. This involves using the appropriate geometric formulas or methods suitable for polygons, particularly for regular decagons where all sides and angles are equal.

Answer

$A \approx 7.69425 s^2$
Answer for screen readers

The area of a regular decagon can be expressed as $A \approx 7.69425 s^2$, where $s$ is the length of one side.

Steps to Solve

  1. Identify the Formula for Area of a Regular Decagon

For a regular decagon, the area can be calculated using the formula:

$$ A = \frac{5}{2} s^2 \cot\left(\frac{\pi}{10}\right) $$

Where $s$ is the length of one side of the decagon.

  1. Calculate the Value of $\cot\left(\frac{\pi}{10}\right)$

To proceed, calculate the value of $\cot\left(\frac{\pi}{10}\right)$.

$$ \cot\left(\frac{\pi}{10}\right) = \frac{1}{\tan\left(\frac{\pi}{10}\right)} $$

Using a calculator or trigonometric tables, we get:

$$ \cot\left(\frac{\pi}{10}\right) \approx 3.0777 $$

  1. Substitute Side Length into the Area Formula

Once we have the value for $s$, substitute it into the area formula along with the calculated value of $\cot\left(\frac{\pi}{10}\right)$ to find the area:

$$ A = \frac{5}{2} s^2 (3.0777) $$

  1. Final Calculation for Area

Multiply and simplify to find the final area:

$$ A \approx 7.69425 s^2 $$

Thus, if you have a specific side length $s$, you can plug it in to find the area of your decagon.

The area of a regular decagon can be expressed as $A \approx 7.69425 s^2$, where $s$ is the length of one side.

More Information

The formula for the area of a decagon stems from its symmetry and geometric properties. Regular polygons often have area formulas that can be derived from trigonometric identities, highlighting the connection between geometry and trigonometry.

Tips

  • Confusing the area formula for regular decagons with those for irregular decagons. Always use the formula appropriate for regular polygons.
  • Not converting angles in degrees if using a calculator that requires it, as some calculators may default to radians.
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