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A decagon is a polygon with ten sides and ten angles. The term decagon comes from the Greek words “deka,” meaning ten, and “gonia,” meaning angle. In geometry, polygons are classified based on the number of sides they have, and a decagon falls under the category of a regular polygon.

A regular decagon is a polygon with ten equal sides and angles. Each angle in a regular decagon measures 144 degrees, and the sum of the interior angles in a decagon is 1440 degrees. The formula to calculate the interior angle of a regular decagon is:

Interior angle = [(n-2) x 180] / n, where n is the number of sides

So for a decagon, the interior angle would be:

Interior angle = [(10-2) x 180] / 10 = 144 degrees

A decagon can be inscribed in a circle, which means all ten vertices of the decagon lie on the circumference of a circle. The radius of the circle that can be inscribed in a decagon is given by the formula:

r = a / (2 x sin(?/10)), where a is the length of one side of the decagon

So, for a regular decagon with a side length of 1 unit, the radius of the circle that can be inscribed in it would be:

r = 1 / (2 x sin(?/10)) = 0.618

Now, let’s take a look at some examples of where decagons can be found:

• Stop signs: One of the most common examples of a decagon is a stop sign. The red octagonal shape with white letters spelling out “STOP” is instantly recognizable and used throughout the world as a universal symbol to bring vehicles to a stop.

• Soccer ball: A soccer ball is made up of 20 hexagons and 12 pentagons, arranged in a pattern known as a truncated icosahedron. Each pentagon in a soccer ball is surrounded by five hexagons, and each hexagon is surrounded by three pentagons and three hexagons. The pentagons in a soccer ball are regular, with all sides and angles being equal, and are essentially decagons with five of their sides replaced with triangles.

• United States Capitol: The dome of the United States Capitol building in Washington, D.C., features a decagonal base. The dome was constructed between 1855 and 1866 and is made of cast iron. It stands 288 feet tall and is one of the most recognizable landmarks in the United States.

• Jewelry: Many jewelry designers use decagons in their designs, particularly for earrings and pendants. The regular shape of the decagon makes it easy to incorporate into geometric designs, and the symmetry of the shape is visually appealing.

• Architecture: Decagons can be found in many types of architecture, particularly in religious buildings. For example, the Dome of the Rock in Jerusalem, which is one of the holiest sites in Islam, features a decagonal shape. The dome was built in the 7th century and is made of wood and brass.

In conclusion, decagons are fascinating geometric shapes that can be found in a variety of applications, from stop signs to soccer balls to religious buildings. Understanding the properties and characteristics of decagons can help us appreciate their beauty and significance in the world around us.

1. What is a decagon?
2. What is the formula to calculate the interior angle of a regular decagon?
3. What is the measure of each angle in a regular decagon?
4. What is the sum of the interior angles in a decagon?
5. What is the formula for the radius of the circle that can be inscribed in a regular decagon?
6. What is the radius of the circle that can be inscribed in a regular decagon with a side length of 1 unit?
7. What are some examples of where decagons can be found?
8. How many hexagons and pentagons are there in a soccer ball?
9. What is the pattern of the hexagons and pentagons in a soccer ball called?
10. What is the Dome of the Rock, and what shape does it feature?

Answers:

1. A polygon with ten sides and ten angles.
2. [(n-2) x 180] / n, where n is the number of sides.
3. Each angle in a regular decagon measures 144 degrees.
4. The sum of the interior angles in a decagon is 1440 degrees.
5. r = a / (2 x sin(?/10)), where a is the length of one side of the decagon.
6. The radius of the circle that can be inscribed in a regular decagon with a side length of 1 unit is 0.618.
7. Stop signs, soccer balls, the United States Capitol, jewelry, and architecture.
8. A soccer ball is made up of 20 hexagons and 12 pentagons.
9. The pattern of the hexagons and pentagons in a soccer ball is called a truncated icosahedron.
10. The Dome of the Rock is one of the holiest sites in Islam and features a decagonal shape.