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Difficulty: Very hardQuadrilaterals and Polygons

The sum of the measures of all interior angles of a convex polygon, excluding one interior angle θ\theta, is equal to 2190∘2190^\circ. If the degree measure of θ\theta is an integer, what is the perimeter of a regular polygon with nn sides, where nn is the number of sides of the original polygon and each side length is θ10\frac{\theta}{10} units?

  1. A
    195
  2. B
    210
  3. 225Answer
  4. D
    240
  5. E
    255

Answer

The perimeter of the regular polygon is 225.
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. Adding the excluded interior angle θ\theta to 2190∘2190^\circ yields the total sum SS. Because 0∘<θ<180∘0^\circ < \theta < 180^\circ, SS must fall strictly between 2190∘2190^\circ and 2370∘2370^\circ. The only multiple of 180∘180^\circ in this interval is 2340∘2340^\circ. Setting (n−2)×180∘=2340∘(n-2)\times 180^\circ = 2340^\circ yields n=15n = 15. Solving for θ\theta gives θ=2340∘−2190∘=150∘\theta = 2340^\circ - 2190^\circ = 150^\circ. The side length is 15010=15\frac{150}{10} = 15, making the perimeter 15×15=22515 \times 15 = 225.

Step-by-Step Solution

1
Set up the inequality for the sum of interior angles of a convex polygon.
The total sum of interior angles for a convex nn-gon is S=(n−2)×180∘S = (n-2) \times 180^\circ. Given S−θ=2190∘S - \theta = 2190^\circ, we have S=2190∘+θS = 2190^\circ + \theta.
The sum of interior angles of any convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
2
Determine the value of SS using the bounds for an interior angle of a convex polygon.
Since 0∘<θ<180∘0^\circ < \theta < 180^\circ, it follows that 2190∘<S<2190∘+180∘=2370∘2190^\circ < S < 2190^\circ + 180^\circ = 2370^\circ. The only multiple of 180∘180^\circ in this range is 2340∘2340^\circ.
SS must be an integer multiple of 180∘180^\circ and θ\theta must be strictly between 0∘0^\circ and 180∘180^\circ.
3
Calculate the number of sides nn and the missing angle θ\theta.
(n−2)×180∘=2340∘  ⟹  n−2=13  ⟹  n=15(n-2) \times 180^\circ = 2340^\circ \implies n - 2 = 13 \implies n = 15. Then θ=2340∘−2190∘=150∘\theta = 2340^\circ - 2190^\circ = 150^\circ.
Solving the linear equations gives exact values for the number of sides and the excluded angle.
4
Compute the perimeter of the regular regular nn-gon.
Side length =θ10=15010=15= \frac{\theta}{10} = \frac{150}{10} = 15. Perimeter =n×side length=15×15=225= n \times \text{side length} = 15 \times 15 = 225.
The perimeter of a regular polygon is the product of its number of sides and its individual side length.

Key Concept

Sum of Interior Angles of Convex Polygons
Estimated Time:2m 0s
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Topics

Quadrilaterals and Polygons

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