What is the number of the side of the polygon if the sum of the measures of the angles of a convex polygon is 900?
The correct answer and explanation is :
The formula for the sum of the interior angles of a convex polygon is given by:
[
\text{Sum of interior angles} = (n – 2) \times 180
]
where ( n ) is the number of sides of the polygon.
Given that the sum of the interior angles of the polygon is 900 degrees, we can set up the equation:
[
(n – 2) \times 180 = 900
]
Now, solve for ( n ):
- First, divide both sides of the equation by 180 to isolate ( n – 2 ):
[
n – 2 = \frac{900}{180}
]
[
n – 2 = 5
]
- Next, add 2 to both sides to solve for ( n ):
[
n = 5 + 2
]
[
n = 7
]
Thus, the polygon has 7 sides.
Explanation:
To understand this, let’s break it down further. The sum of the interior angles of a polygon can be derived from the fact that any polygon can be divided into triangles. A triangle has a total interior angle sum of 180 degrees, and by dividing the polygon into triangles, we get the formula ( (n – 2) \times 180 ) for the sum of the interior angles, where ( n ) is the number of sides.
For example, a quadrilateral (4-sided polygon) can be divided into 2 triangles, so the sum of the interior angles is ( (4 – 2) \times 180 = 360 ) degrees. A pentagon (5-sided polygon) can be divided into 3 triangles, so the sum of the interior angles is ( (5 – 2) \times 180 = 540 ) degrees, and so on.
In this problem, the sum of the interior angles is 900 degrees, which means the polygon must have 7 sides.