what is the sum of the interior angles of a decagon
The correct answer and explanation is:
The sum of the interior angles of a decagon is 1440 degrees.
Explanation:
A decagon is a polygon with 10 sides. To find the sum of the interior angles of any polygon, use the formula:
Sum of interior angles = (n – 2) × 180 degrees
where “n” is the number of sides of the polygon.
For a decagon, n = 10. Substituting this into the formula:
Sum = (10 – 2) × 180
Sum = 8 × 180
Sum = 1440 degrees
This means that if you add up all the interior angles inside a decagon, the total will be 1440 degrees.
The reason behind this formula lies in the fact that any polygon can be divided into triangles by drawing non-overlapping diagonals from one vertex. Each triangle has an angle sum of 180 degrees. A polygon with n sides can be divided into (n – 2) triangles. Multiplying the number of triangles by 180 degrees gives the total interior angle sum of the polygon.
For example, for a triangle (3 sides), the sum of interior angles is (3 – 2) × 180 = 1 × 180 = 180 degrees, which is well known. For a quadrilateral (4 sides), it is (4 – 2) × 180 = 2 × 180 = 360 degrees.
Applying this logic to a decagon, dividing it into 8 triangles explains why the total is 1440 degrees. This property helps in many geometric calculations, including finding the measure of each interior angle if the polygon is regular (all sides and angles equal). For a regular decagon, each interior angle is 1440 ÷ 10 = 144 degrees.
Understanding this concept provides a foundation for studying polygons and their properties in geometry.