How many sides are in polygon if each interior angle is 165 degrees?
The correct answer and explanation is :
To find the number of sides in a polygon when the interior angle is 165 degrees, we use the formula for the interior angle of a regular polygon. The formula for the interior angle of a regular polygon is:
[
\text{Interior angle} = \frac{(n – 2) \times 180^\circ}{n}
]
Where:
- ( n ) is the number of sides of the polygon.
- ( 180^\circ ) represents the sum of interior angles of a triangle (since every polygon can be broken into triangles).
- ( (n – 2) ) is the number of triangles that can be formed from the polygon.
We are given that the interior angle is 165 degrees. Now, we need to find ( n ) (the number of sides of the polygon).
Step 1: Set up the equation
We know the interior angle is 165 degrees, so we substitute this value into the formula:
[
165 = \frac{(n – 2) \times 180}{n}
]
Step 2: Solve for ( n )
First, multiply both sides of the equation by ( n ) to get rid of the denominator:
[
165n = (n – 2) \times 180
]
Now expand the right side:
[
165n = 180n – 360
]
Next, subtract ( 165n ) from both sides:
[
0 = 15n – 360
]
Add 360 to both sides:
[
360 = 15n
]
Now, divide both sides by 15:
[
n = \frac{360}{15} = 24
]
Conclusion:
The polygon has 24 sides.
Explanation:
A polygon’s interior angle depends on the number of sides it has. Using the formula for the interior angle, we can calculate the number of sides when the angle is known. By solving the equation, we find that a regular polygon with an interior angle of 165 degrees must have 24 sides. This process shows how the relationship between the number of sides and the measure of the interior angle can be used to solve for unknown values in geometric problems.