SOLUTION: What is the sum of the measures of the interior angles of a regular polygon if each exterior angle measures 90?

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Question 262803: What is the sum of the measures of the interior angles of a regular polygon if each exterior angle measures 90?

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the sum of the exterior angles of a polygon is always equal to 360 degrees.

the number of sides of the polygon is always equal to 360 / by the exterior angle.

a regular polygon is assumed (equal sides and equal angles all around).

each exterior angle of a polygon is the supplement of its corresponding interior angle.

the sum of the interior angles of a polygon is equal to (n-2)*180.

it is also equal to the number of sides times each interior angle.


first example:

sum of the interior angles of a regular triangle is (3-2) * 180 = 1 * 180 = 180.

each interior angle of the triangle = 180/3 = 60 degrees.

each exterior angle of the triangle = 180 - 60 = 120 degrees.

360 / 120 = 3 which is the number of sides of the triangle based on the exterior angle.

next example:

sum of the interior angles of a regular quadrilateral = (4-2) * 180 = 2 * 180 = 360.

each interior angle of the quadrilateral = 90 degrees.

each exterior angle of the quadrilateral = 180 - 90 = 90 degrees.

number of sides of the quaqdrilateral = 360 / 90 = 4 based on the exterior angles.

next example:

sum of the interior angles of a regular pentagon = (5-2) * 180 = 3 * 180 = 540.

each interior angle of the pentagon = 540/5 = 108

each exterior angle of the pentagon = 180 - 108 = 72.

the number of sides of the pentagon = 360 / 72 = 5 based on the exterior angle.

last example:

sum of the interior angles of a regular hexagon = (6-2) * 180 = 4 * 180 = 720.

each interior angle of the pentagon = 720/6 = 120.

each exterior angle of the pentagon = 180 - 120 = 60.

the number of sides of the hexagon = 360/60 = 6 based on the exterior angle.

in your problem:

you are given that the exterior angle = 90 degrees.

take 360 and divide it by 90 and you get 4 sides to the polygon.

each interior angle of the polygon = 180 - 90 = 90 degrees.

4 times 90 = 360 degrees.

the sum of the interior angles of the polygon = 360 degrees.

the polygon is a quadrilateral.

the sum of the interior angles of a quadrilateral = (4-2) * 180 = 2 * 180 = 360.

your answer is:

the sum of the interior angles of a polygon = 360 if each exterior angle = 90 degrees.