Answer:
- 72 sides
- interior angle: 175°
- exterior angle: 5°
- exterior angle total: 360°
Explanation:
Given a regular polygon whose interior angles total 12,600 degrees, you want to know the number of sides, the measures of each interior and exterior angle, and the sum of the exterior angles.
Angle sum
The sum of angles of a convex polygon is given by the formula ...
angle sum = 180° × (n -2)
where n is the number of sides.
Application
Substituting the given angle sum, we can solve for n:
12,600 = 180(n -2)
70 = n -2 . . . . . divide by 180
72 = n . . . . . . . add 2
The regular polygon has 72 sides.
Interior angle
The interior angles of a regular polygon are congruent, so each one is ...
12,600°/72 = 175°
Each interior angle = 175°.
Exterior angle
The exterior angle at a vertex is the supplement of the interior angle there:
180° -175° = 5°
Each exterior angle = 5°.
Sum of exterior angles
The 72 exterior 5° angles have a total of ...
72 × 5° = 360°
Sum of the exterior angles = 360°.
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Additional comment
The invariant with convex polygons, regular or not, is that the sum of exterior angles is 360°. This fact is what gives rise to the formula for interior angles.