Examkin

Question

Difficulty: MediumAngles, Parallel Lines, and Polygons

The sum of the interior angles of a convex polygon is three times the sum of its exterior angles. Calculate the number of sides of the polygon.

Answer: 8

Answer

The polygon has 8 sides.
The sum of the exterior angles of any convex polygon is always 360∘360^\circ. Since the sum of the interior angles is three times this value, the interior angle sum is 3×360∘=1080∘3 \times 360^\circ = 1080^\circ. Setting (n−2)×180∘=1080∘(n - 2) \times 180^\circ = 1080^\circ gives n−2=6n - 2 = 6, which yields n=8n = 8 sides.

Step-by-Step Solution

1
Determine the exterior angle sum
The sum of exterior angles for any convex polygon is 360∘360^\circ.
The exterior angles of any convex polygon sum to a full turn (360∘360^\circ).
2
Compute the sum of the interior angles
Sum of interior angles = 3×360∘=1080∘3 \times 360^\circ = 1080^\circ.
The question states that the interior angle sum is three times the exterior angle sum.
3
Apply the interior angle sum formula
(n−2)×180∘=1080∘(n - 2) \times 180^\circ = 1080^\circ.
The interior angle sum of an nn-sided convex polygon is given by (n−2)×180∘(n - 2) \times 180^\circ.
4
Solve for nn
n−2=1080∘180∘=6  ⟹  n=8n - 2 = \frac{1080^\circ}{180^\circ} = 6 \implies n = 8.
Dividing 1080∘1080^\circ by 180∘180^\circ gives 66, and adding 22 gives n=8n = 8.

Key Concept

Relationship between the sum of interior and exterior angles of a convex polygon
Estimated Time:1m 0s
Rate this question

Topics

Angles, Parallel Lines, and Polygons

More questions from this topic