if an exterior angle of a regular polygon is 45° then find the number of sides.
Answers
Answer:
Please see diagram.
A regular polygon has n sides and n vertices. The lines joining the vertices and the center O of polygon create n isosceles triangles. The side of the polygon becomes base of these triangles. The angle at the center in the triangle is Ф = 360°/n.
As the two angles at the base are A/2 = (180° - Ф )/2.
The interior angle at a vertex is A = 180°- Ф.
So the exterior angle is = Ф = 360°/n = 2π/n
So if 360°/n = 45°, n = 360°/45 = 8
It is a regular octagon with 8 sides.
The number of line connecting each vertex to another is : ⁸C₂ = 8 * 7 /2 = 28.
Of these, there are 8 sides among adjacent vertices.
The remaining are the diagonals and are 28 – 8 = 20.
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n >= 3
Formula for number of diagonals = n(n-1)/2 - n = n(n-3)/2.
Sum total of all exterior angles = 360° = 2 π for any regular polygon.
One exterior angle = 360° / n = 2π/n
one interior angle = 180° – 360°/n = 180° (n-2)/n = (n-2)π/n
Angle made by a side at the cente = 360°/n = 2π/n
Sum total of all interior angles = n * 180° (n-2)/n = 180° (n-2) = (n-2)π
