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Difficulty: MediumPolygon Angles and Properties

A convex polygon has nn sides. The sum of the measures of its interior angles is 66 times the sum of the measures of its exterior angles (one at each vertex). What is the value of nn?

Answer: 14

Answer

The number of sides, nn, of the convex polygon is 14.
The sum of the interior angles of a convex polygon with nn sides is given by the formula (n−2)×180∘(n-2) \times 180^\circ, and the sum of its exterior angles is always 360∘360^\circ. According to the problem, the sum of the interior angles is 66 times the sum of the exterior angles, yielding the equation (n−2)×180=6×360(n - 2) \times 180 = 6 \times 360. Dividing both sides of the equation by 180180 gives n−2=12n - 2 = 12. Adding 22 to both sides results in n=14n = 14.

Step-by-Step Solution

1
State the sum of interior and exterior angles formulas.
Interior sum = (n−2)×180∘(n-2) \times 180^\circ, Exterior sum = 360∘360^\circ.
To represent the geometric properties of the polygon algebraically.
2
Set up the equation based on the given ratio.
(n−2)×180=6×360(n-2) \times 180 = 6 \times 360.
The problem states the interior sum is 6 times the exterior sum.
3
Solve the equation for nn.
n=14n = 14.
Divide by 180 to get n−2=12n - 2 = 12, then add 2 to both sides.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ, and the sum of the exterior angles (one per vertex) is always 360∘360^\circ.
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Topics

Polygon Angles and Properties

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