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Difficulty: HardPolygon Angles and Properties

In a convex polygon, the measures of the interior angles form an arithmetic progression. The smallest interior angle measures 120∘120^\circ, and the common difference between consecutive interior angles is 5∘5^\circ. What is the number of sides of this polygon?

Answer: 9

Answer

The number of sides of the polygon is 9.
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. Since the angles form an arithmetic progression with the first term a=120∘a = 120^\circ and common difference d=5∘d = 5^\circ, their sum is also given by the arithmetic series formula: Sn=n2[2(120)+(n−1)5]=n2(5n+235)S_n = \frac{n}{2}[2(120) + (n-1)5] = \frac{n}{2}(5n + 235). Setting the two sums equal yields n2(5n+235)=180(n−2)\frac{n}{2}(5n + 235) = 180(n - 2), which simplifies to the quadratic equation n2−25n+144=0n^2 - 25n + 144 = 0. Solving this gives n=9n = 9 or n=16n = 16. Because the polygon is convex, every interior angle must be less than 180∘180^\circ. If n=16n = 16, the largest angle would be 120∘+15(5∘)=195∘120^\circ + 15(5^\circ) = 195^\circ, which is impossible for a convex polygon. If n=9n = 9, the largest angle is 120∘+8(5∘)=160∘120^\circ + 8(5^\circ) = 160^\circ, which is valid. Therefore, the number of sides must be 9.

Step-by-Step Solution

1
Set up the equation equating the geometric sum of interior angles to the arithmetic series sum.
The sum of the interior angles of a convex nn-gon is (n−2)×180∘(n - 2) \times 180^\circ. The sum of the arithmetic sequence of angles is Sn=n2[2(120∘)+(n−1)5∘]S_n = \frac{n}{2}[2(120^\circ) + (n - 1)5^\circ]. Setting them equal gives: n2(5n+235)=180(n−2)\frac{n}{2}(5n + 235) = 180(n - 2).
This establishes the algebraic relationship between the polygon's geometric properties and the given sequence of angle measures.
2
Simplify the equation and solve the resulting quadratic equation for nn.
5n2+235n=360n−720⇒5n2−125n+720=0⇒n2−25n+144=05n^2 + 235n = 360n - 720 \Rightarrow 5n^2 - 125n + 720 = 0 \Rightarrow n^2 - 25n + 144 = 0. Factoring the quadratic yields (n−9)(n−16)=0(n - 9)(n - 16) = 0, so n=9n = 9 or n=16n = 16.
Solving the quadratic equation yields all mathematically possible values for the number of sides.
3
Apply the convexity constraint to determine the valid number of sides.
For a polygon to be convex, every interior angle must be less than 180∘180^\circ. For n=16n = 16, the largest angle is 120∘+15(5∘)=195∘120^\circ + 15(5^\circ) = 195^\circ, which is impossible. For n=9n = 9, the largest angle is 120∘+8(5∘)=160∘120^\circ + 8(5^\circ) = 160^\circ, which is valid.
The definition of a convex polygon requires all interior angles to be strictly less than 180∘180^\circ, which eliminates the extraneous solution of 16.

Key Concept

Sum of interior angles of a convex polygon and arithmetic progressions
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Topics

Polygon Angles and Properties

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