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Chapter 3: Problem 5

For each polygon, find (a) the interior angle sum and (b) the exterior angle sum. Decagon

Short Answer

Expert verified
The interior angle sum of a decagon is \(1440\) degrees and the exterior angle sum is \(360\) degrees.

Step by step solution

01

Find the Interior Angle Sum

The formula to find the sum of the angles in a polygon is: \((n-2) * 180\), where n is the number of sides of the polygon. In this exercise, you have a decagon, which has 10 sides. Thus, for the decagon, plug 10 into the formula to get: \((10-2) * 180 = 1440\). Thus the interior angle sum of a decagon is \(1440\) degrees.
02

Find the Exterior Angle Sum

The exterior angle sum of any polygon is always \(360\) degrees. Therefore the exterior angle sum of a decagon is also \(360\) degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interior Angle Sum
The interior angle sum of any polygon is the total degrees of all the inside angles. Understanding this concept is fundamental in geometry.
The formula for finding the interior angle sum is simple:
  • It鈥檚 \( (n-2) \times 180\), where \( n \) is the number of sides of the polygon.
  • This formula arises because you can divide any polygon into \( n-2 \) triangles.
  • Each triangle has an interior angle sum of \( 180\) degrees.
For a decagon, which is a ten-sided polygon, substitute \( n=10 \) into the formula:
  • Calculation: \( (10-2) \times 180 = 1440 \) degrees.
So, the interior angle sum of a decagon is \( 1440 \) degrees. This concept helps to not only find the sum quickly but also verify the shape's structure in geometric problems.
Exterior Angle Sum
The exterior angles of a polygon are the angles formed between one side of the polygon and the extension of an adjacent side. Understanding their sum is crucial and universal in geometry.
The formula to find the sum of exterior angles is consistently easy, as follows:
  • The sum of the exterior angles of any polygon, regardless of the number of sides, is always \( 360 \) degrees.
  • This is because, pieced together, these angles complete a full circle around the polygon.
For a decagon, this characteristic rule applies identically:
  • The exterior angle sum is \( 360 \) degrees.
While it may seem surprising at first, this uniformity is due to the nature of polygons, where extending the sides consistently results in circular paths.
Decagon
A decagon is a type of polygon with some specific properties that make it unique among shapes.
Getting to know a decagon involves understanding its structure:
  • A decagon has \( 10 \) sides.
  • It can be regular or irregular. In a regular decagon, all sides and angles are equal.
  • Each interior angle in a regular decagon is \( 144 \) degrees, because you divide the interior angle sum (\( 1440 \) degrees) by \( 10 \).
  • Similarly, each exterior angle in a regular decagon is \( 36 \) degrees, calculated by dividing the exterior angle sum (\( 360 \) degrees) by \( 10 \).
Decagons are not just theoretical shapes but also appear in artistry and architecture. Their symmetrical properties make them visually appealing and structurally balanced for various designs.

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