# If the exterior angle of a regular polygon is 45 degrees, then what is the no. of sides in the polygon and the no. of diagonals?

I need this answer with the steps and formulas used

2
by Sam1142 07.10.2014

Log in to add a comment

I need this answer with the steps and formulas used

2
by Sam1142 07.10.2014

Log in to add a comment

The Brainliest Answer!

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

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.

=========================

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

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.

=========================

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