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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 33 times the sum of the measures of its exterior angles. What is the value of nn?

Answer: 8

Answer

8
The sum of the interior angle measures of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ. The sum of the exterior angle measures is always 360∘360^\circ. According to the problem, the sum of the interior angles is 33 times the sum of the exterior angles, which can be written as the equation (n−2)×180=3×360(n-2) \times 180 = 3 \times 360. Simplifying the right side gives (n−2)×180=1080(n-2) \times 180 = 1080. Dividing both sides by 180180 results in n−2=6n - 2 = 6. Adding 22 to both sides yields n=8n = 8.

Step-by-Step Solution

1
Use the formula for the sum of the interior angle measures of a convex polygon with nn sides.
The sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ.
By the polygon interior angle sum theorem, the sum of the interior angles of any convex nn-gon is (n−2)×180∘(n-2) \times 180^\circ.
2
Identify the sum of the exterior angle measures of a convex polygon.
The sum of the exterior angles is 360∘360^\circ.
The sum of the exterior angles of any convex polygon is always constant and equals 360∘360^\circ, regardless of the number of sides.
3
Set up the equation relating the two sums as given in the problem statement.
(n−2)×180∘=3×360∘(n-2) \times 180^\circ = 3 \times 360^\circ
The problem states that the sum of the interior angles is 33 times the sum of the exterior angles.
4
Solve the equation for nn.
n=8n = 8
Divide both sides by 180∘180^\circ to get n−2=6n - 2 = 6, then add 22 to both sides to find n=8n = 8.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and the sum of its exterior angles is 360∘360^\circ.
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Topics

Polygon Angles and Properties

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