A regular polygon has an interior angle which less than ten times its exterior angle by 84°.Find the sum of all its interior angle?
let the number of sides be n
each interior angle = 180(n-2)/n
each exterior angle = 360/n
180(n-2)/n < 10(360/n) by 84
180(n-2)/n = 10(360/n) - 84
times n, which is positive
180(n-2) = 3600 - 84n
180n - 360 = 3600 - 84n
264n = 3960
n = 15
sum of angles = 180(15-2)° = 2340°
check my arithmetic
If the number of sides is n, then we have
180(n-2)/n = 10(360/n)-84
n=15
So, the sum of its interior angles is 180(15-2) = 2340°
To find the sum of all the interior angles of a regular polygon, we need to know the number of sides (n) of the polygon. Let's denote the measure of each interior angle as "x" degrees.
We are given that the interior angle is less than ten times the exterior angle by 84°. The exterior angle of any polygon is the supplement of the interior angle. In other words, the exterior angle is 180° - x.
According to the given information, we can set up the equation:
x = 10(180° - x) - 84°
Let's solve this equation:
x = 1800° - 10x - 84°
11x = 1800° - 84°
11x = 1716°
x = 156°
The measure of each interior angle is 156°.
Now, we can find the sum of all the interior angles of the polygon using the formula:
Sum = (n - 2) * 180°
Substituting the value of x into the formula, we have:
Sum = (n - 2) * 180°
Sum = (n - 2) * 180° = (n - 2) * 156°
Hence, the sum of all the interior angles of the regular polygon is (n - 2) * 156°.