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Difficulty: EasyPolygon Angles and Properties

The ratio of the measure of an interior angle of a regular polygon to the measure of its exterior angle is 3:13:1. How many sides does this polygon have?

  1. A
    4
  2. B
    6
  3. 8Answer
  4. D
    10
  5. E
    12

Answer

8
The interior angle and exterior angle of a polygon at any vertex are supplementary, meaning they add up to 180∘180^\circ. Given the ratio of the interior angle to the exterior angle is 3:13:1, we can express their measures as 3x3x and xx. Solving the equation 3x+x=180∘3x + x = 180^\circ gives 4x=180∘4x = 180^\circ, which means x=45∘x = 45^\circ. Therefore, the measure of each exterior angle of the regular polygon is 45∘45^\circ. Since the sum of the exterior angles of any convex polygon is always 360∘360^\circ, the number of sides nn is calculated by dividing 360∘360^\circ by the measure of one exterior angle: n=360∘45∘=8n = \frac{360^\circ}{45^\circ} = 8.

Step-by-Step Solution

1
Set up an equation for the interior and exterior angles using the given ratio.
Let the measure of the exterior angle be xx and the measure of the interior angle be 3x3x.
The ratio of the interior angle to the exterior angle is 3:13:1, so their measures can be represented as 3x3x and xx respectively.
2
Use the fact that the interior angle and exterior angle at any vertex of a polygon are supplementary (form a linear pair).
3x+x=180∘  ⟹  4x=180∘  ⟹  x=45∘3x + x = 180^\circ \implies 4x = 180^\circ \implies x = 45^\circ.
An interior angle and its adjacent exterior angle always lie on a straight line and sum to 180∘180^\circ.
3
Calculate the number of sides of the regular polygon using the measure of one exterior angle.
n=360∘45∘=8n = \frac{360^\circ}{45^\circ} = 8.
The sum of the exterior angles of any convex polygon is always 360∘360^\circ. For a regular polygon with nn sides, each exterior angle measures 360∘n\frac{360^\circ}{n}.

Key Concept

The relationship between the interior and exterior angles of a regular polygon, and the formula relating the number of sides to the sum of the exterior angles.
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Topics

Polygon Angles and Properties

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