The interior angles of a polygon are in AP. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.What is the answer?

Question

Answer

Answer

AP is 120,125,130…. An=n/2[2*120 (n-1)5]=180(n-2)n*n-16n-9n 144=0 n(n-16)-9(n-16)=0 n=16 and n=9

Answer

Given:
- The interior angles of a polygon are in arithmetic progression (AP).
- The smallest angle is 120°.
- The common difference is 5°.

To Find:
The number of sides of the polygon.

Solution:

Step 1: Finding the Common Difference:
The common difference between the interior angles of the polygon is given as 5°. Let's denote it as 'd'.

Therefore, d = 5°.

Step 2: Finding the Sum of Interior Angles:
The sum of interior angles of any polygon can be found using the formula:
Sum = (n - 2) * 180°, where 'n' is the number of sides of the polygon.

Since we know the smallest angle is 120°, we can find the sum of interior angles using the formula for the first term 'a' and the common difference 'd' as follows:

a = 120° (smallest angle)
d = 5° (common difference)

Using the formula for the sum of an arithmetic series:
Sum = (n/2) * (2a + (n-1)d)

Replacing the values:
(n/2) * (2 * 120° + (n-1) * 5°) = (n - 2) * 180°

Simplifying the equation:
(n/2) * (240° + 5n - 5°) = (n - 2) * 180°
(n/2) * (235° + 5n) = (n - 2) * 180°
235n + 5n^2 = 360n - 720
5n^2 - 125n + 720 = 0

Solving this quadratic equation, we get two values for 'n': n = 9 and n = 16.

Step 3: Checking the Validity:
Since a polygon cannot have less than 3 sides, we can ignore n = 9. Therefore, the number of sides of the polygon is 16.

Answer:
The number of sides of the polygon is 16.

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