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The Fascinating World of Regular Decagons

CEO Khai Intela
Did you know that a decagon is a polygon with ten sides and ten vertices? It's a shape that can be both convex and concave. But what makes a decagon truly special is when it's...

Did you know that a decagon is a polygon with ten sides and ten vertices? It's a shape that can be both convex and concave. But what makes a decagon truly special is when it's regular, with equal sides and equal interior angles. In this article, we'll delve into the theoretical background and explore the properties, construction, and formulas related to regular decagons. So, let's dive in!

Understanding Regular Decagons

A regular decagon has some interesting properties that set it apart from other polygons. For example, it has ten axes of symmetry, some passing through diagonally opposite vertices and others through the midpoints of opposite edges. All these axes meet at a common point, the center of the decagon.

The interior angles of a regular decagon are equal and each measures 144 degrees. The central angle, which is the angle between two successive diagonals, measures 36 degrees. The interior angles and central angles are supplementary, meaning their sum is always 180 degrees.

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Regular decagons also have two important circles associated with them. The circumcircle is a circle that passes through all ten vertices of the decagon. Its center is also the center of the decagon. The circumradius, which is the radius of the circumcircle, is related to the length of the decagon's sides.

On the other hand, the incircle is a circle that passes through the midpoints of all ten edges of the decagon. It is tangent to all ten edges and shares the same center with the circumcircle. The inradius is the radius of the incircle.

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Calculating Area and Perimeter

The area of a regular decagon can be calculated by dividing it into ten identical isosceles triangles. The height of each triangle is equal to the inradius, and the area of each triangle is half the product of the decagon's side length and the inradius. Therefore, the total area of the decagon can be approximated as 7.694 times the square of the side length.

The perimeter of a regular decagon is simply the sum of the lengths of all its sides. For a regular decagon, with ten sides, the perimeter is equal to ten times the side length.

Constructing a Regular Decagon

A regular decagon can be constructed using only a ruler and a compass, given its circumradius. The construction involves drawing a series of arcs and lines to locate the ten vertices of the decagon. The detailed step-by-step procedure can be seen in the figure below.

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Quick Reference Table

For your convenience, here's a concise list of the main formulas and approximations related to regular decagons:

  • Circumradius (Rc) = 1.618 times the side length (a)
  • Inradius (Ri) = 1.539 times the side length (a)
  • Height (h) = 3.078 times the side length (a)
  • Width (w) = 3.236 times the side length (a)
  • Area (A) = 7.694 times the square of the side length (a)

With these formulas and approximations at your fingertips, you'll be able to solve various problems and explore the fascinating world of regular decagons!

So, the next time you encounter a regular decagon, remember its unique properties, construction techniques, and the formulas that define it. Regular decagons are not only mathematically intriguing but also visually captivating. Happy exploring!

Please note that all images used in this article are sourced from saigonintela.vn and are for educational purposes only.

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