A regular polygon with the minimum number of sides is constructed such that each internal angle is an obtuse angle how many diagonals would this polygon have
Answers
Given,
The regular polygon (with minimum number of sides) has obtuse angles as the internal angles.
To find,
The number of diagonals of that polygon.
Solution,
We can simply solve this mathematical problem by using the following mathematical process.
Now, we have to find out the number of sides of that polygon, with the help of the given conditions.
Formula used :
Value of each internal angles of a regular polygon with "n" sides = (n-2)×180°/n
So,
1) A equilateral triangle (n=3) has internal angles of 60°. [(n-2)×180°/n = (3-2)×180°/3 = 60°]
2) A square (four sided regular polygon or n=4) has internal angles of 90°. [(n-2)×180°/n = (4-2)×180°/4 = 90°]
3) A regular pentagon (n=5) has internal angles of 108°.[(n-2)×180°/n = (5-2)×180°/5 = 108°]
So, the regular Pentagon is the first regular polygon in the above mentioned series, which has an obtuse angle as the internal angle. (108° is greater than 90° but less than 180°)
That's why, we can come to a conclusion that the regular pentagon is the polygon with minimum number of sides which has obtuse angles as internal angles.
Now, number of diagonals
= n(n-3)/2
= 5×(5-3)/2 [Pentagon has five sides, that's why n = 5]
= (5×2)/2
= 10/2
= 5
Hence, the number of diagonals is 5