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Difficulty: MediumQuadrilaterals and Polygons

In a regular polygon, the ratio of the measure of an interior angle to the measure of an exterior angle is 7:27:2. What is the total number of diagonals of this polygon?

Answer: 27

Answer

The total number of diagonals of the regular polygon is 2727.
The correct answer is 2727. An interior angle and an exterior angle of a polygon are supplementary (180∘180^\circ). Given the ratio 7:27:2, the exterior angle is 29×180∘=40∘\frac{2}{9} \times 180^\circ = 40^\circ. Since the sum of exterior angles of any convex polygon is 360∘360^\circ, the number of sides is n=360∘40∘=9n = \frac{360^\circ}{40^\circ} = 9. Using the formula for the number of diagonals, n(n−3)2\frac{n(n-3)}{2}, we obtain 9(9−3)2=27\frac{9(9-3)}{2} = 27.

Step-by-Step Solution

1
Determine the measure of the exterior angle using the given interior-to-exterior ratio.
Exterior angle measure = 40∘40^\circ
At any vertex of a polygon, the interior angle and exterior angle sum to 180∘180^\circ. With a ratio of 7:27:2, the exterior angle represents 27+2=29\frac{2}{7+2} = \frac{2}{9} of the total 180∘180^\circ.
2
Calculate the number of sides (nn) of the regular polygon.
n=9n = 9
The sum of the exterior angles of any convex polygon is 360∘360^\circ. Since all exterior angles in a regular polygon are equal, n=360∘40∘=9n = \frac{360^\circ}{40^\circ} = 9.
3
Calculate the total number of diagonals using the formula n(n−3)2\frac{n(n-3)}{2}.
Number of diagonals = 2727
Substituting n=9n = 9 into n(n−3)2\frac{n(n-3)}{2} yields 9×62=27\frac{9 \times 6}{2} = 27.

Key Concept

Interior and exterior angle properties of regular polygons, and the diagonal counting formula for convex polygons.
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Topics

Quadrilaterals and Polygons

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