Question

Find the number of sides of a regular polygon if each of its interior angle is $\sf{(\frac{4\pi}{5})^{c}}$.​

Answer 1

Step-by-step explanation:

Let total number of exterior angles = n

Let sum of all exterior sngles = S

Since, it is a regular polygon, its all exterior angles are equal.

Therefore,

$s = n \times (\frac{4\pi}{5} \times \frac{180}{\pi})$

(Converting $\frac{4\pi}{5}$ into Degrees)

$s = 144n$

Now, by the formula,

$s = (n - 2) \times 180$

Putting the value of S,

$144n = 180n - 360$

$36n = 360$

$n = 10$

Therefore, no. of exterior angles = 10

For any polygon, no. of sides = no. of exterior angles.

Hence, no. of sides of regular polygon = 10

So, the polygon is a Decagon.

Answer 2

Solution:-This question can be simply solved by the formula,

But we will solve it by understanding the concept.....

see the polygon in the given figure

without counting it's side,,let it's total side be n

let the angle formed by sides be x

now,sum of angles=number of sides×value of that angle=n×x

now,here if we count the angle of all the traingle

i.e angleOBC+angleOCB+angleBOC+angleOBA+.....

we can say from here that,in that if we subtract angle(BOC+AOB+AOF+FOE+EOD+DOC),then we will get the sum of interior angle.

and that value which we have to subtract is 360, because these angles form a circle

so,from.here we can say

n×180-360=4×180×n/5. (n×180 is the sum of angle of all the triangles formed)

180n-144n=360

36n=360

n=10

so ,sides will be 10