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Area Del Decágono Regular: Understanding The Formula

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Perímetro y área de polígonos regulares. Aula05mate
Perímetro y área de polígonos regulares. Aula05mate from aula05mate.com

As we delve into the world of geometry, we come across a fascinating shape called the decagon. A decagon is a polygon with ten sides and ten angles. In this article, we will explore the area of a regular decagon and the formula to calculate it.

What is a Regular Decagon?

A regular decagon is a decagon where all the sides and angles are of equal measure. It is a symmetrical shape with ten congruent isosceles triangles. The sum of the interior angles of a decagon is 1440 degrees, and each angle measures 144 degrees.

Formula to Calculate the Area of a Regular Decagon

The formula to calculate the area of a regular decagon is:

  • Area = (5/4) x (s^2) x (√(5 + 2√5))

    • Here, 's' represents the length of each side of the decagon. This formula can be used to calculate the area of any regular decagon, regardless of its size.

    Example:

    Let us consider a regular decagon with a side length of 6 cm. Using the formula mentioned above, we can calculate the area of the decagon as:

  • Area = (5/4) x (6^2) x (√(5 + 2√5))
  • Area = 93.107 cm²

    • Derivation of the Formula

    The formula to calculate the area of a regular decagon can be derived using trigonometry. The decagon can be divided into ten congruent isosceles triangles, each with an apex angle of 36 degrees. By bisecting the triangle, we get two right-angled triangles, as shown in the figure below:

    Decagon Image

    Using trigonometry, we can calculate the height of the isosceles triangle as:

  • tan(18) = (h/2) / (s/2)
  • h = s x tan(18)

    • The area of the isosceles triangle can be calculated as:

  • Area of Triangle = (1/2) x base x height
  • Area of Triangle = (1/2) x s x h
  • Area of Triangle = (1/2) x s x s x tan(18)

    • As there are ten isosceles triangles in a regular decagon, the total area can be calculated as:

  • Area of Decagon = 10 x Area of Triangle
  • Area of Decagon = 5 x s^2 x tan(18)

    • We can simplify this formula further using the trigonometric identity:

  • tan(36) = (√5 - 1) / 2

    • Substituting the value of tan(36) in the formula, we get:

  • Area of Decagon = (5/4) x s^2 x (√(5 + 2√5))

    • Properties of a Regular Decagon

    A regular decagon has several interesting properties that make it unique:

  • It has ten lines of symmetry, passing through each vertex and the midpoint of opposite sides.
  • The diagonals of a decagon are 35 in number, and they intersect at 25 distinct points.
  • The ratio of the diagonals to the sides of a regular decagon is (√5 + 1) / 2.

    • Applications of a Regular Decagon

    The regular decagon has several applications in the field of science, engineering, and art. It is used in the design of buildings, bridges, and other structures. The decagon is also used in the creation of complex geometric patterns and designs.

    Conclusion

    The regular decagon is a fascinating shape with unique properties and applications. The formula to calculate the area of a regular decagon is simple yet elegant, derived using trigonometry. We hope that this article has helped you understand the concept of a regular decagon and its area formula.

    Remember, the key to mastering geometry is practice, so go ahead and try solving some problems involving the regular decagon. Happy learning!

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