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Question

Difficulty: MediumQuadrilaterals and Polygons

The sum of the interior angles of a convex polygon is exactly three times the sum of its exterior angles. How many sides does this polygon have?

Answer: 8

Answer

8
Applying the core polygon properties, the interior angle sum is 180∘(n−2)180^\circ(n - 2) and the exterior angle sum is 360∘360^\circ. Setting 180∘(n−2)=3(360∘)180^\circ(n - 2) = 3(360^\circ) and solving for nn yields 8.

Step-by-Step Solution

1
Identify the interior angle sum formula.
Sum = 180∘(n−2)180^\circ(n - 2)
This formula provides the total sum of all interior angles for an nn-sided polygon.
2
Identify the exterior angle sum property.
Sum = 360∘360^\circ
A fundamental property of all convex polygons is that the sum of their exterior angles is exactly 360∘360^\circ.
3
Set up the algebraic equation.
180∘(n−2)=3×360∘180^\circ(n - 2) = 3 \times 360^\circ
The question states the interior sum is exactly three times the exterior sum.
4
Solve the equation for nn.
n=8n = 8
Dividing 1080∘1080^\circ by 180∘180^\circ yields 6, and adding 2 gives a final side count of 8.

Key Concept

Polygon Angle Sum Theorems
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Topics

Quadrilaterals and Polygons

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