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Difficulty: MediumPolygon Angles and Properties

For a certain convex polygon, the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1. How many sides does this polygon have?

  1. A
    7
  2. B
    8
  3. C
    10
  4. 12Answer
  5. E
    14

Answer

The correct answer is 12 sides.
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n-2) \times 180^\circ. The sum of the exterior angles of any convex polygon is always 360∘360^\circ. Since the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1, we can set up the equation: (n−2)×180∘360∘=51\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}. Simplifying the fraction on the left gives n−22=5\frac{n-2}{2} = 5. Multiplying both sides by 2 gives n−2=10n - 2 = 10, and adding 2 to both sides gives n=12n = 12. Therefore, the polygon has 12 sides.

Step-by-Step Solution

1
Write the formulas for the sum of the interior angles and the sum of the exterior angles of a convex polygon.
The sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ, where nn is the number of sides. The sum of the exterior angles is always 360∘360^\circ for any convex polygon.
These formulas represent the components of the given ratio.
2
Set up the ratio equation using the given information.
(n−2)×180∘360∘=51\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}
The problem states that the ratio of the sum of the interior angles to the sum of the exterior angles is 5:15:1.
3
Simplify the equation and solve for the number of sides, nn.
n−22=5  ⟹  n−2=10  ⟹  n=12\frac{n-2}{2} = 5 \implies n-2 = 10 \implies n = 12
Simplifying 180360\frac{180}{360} to 12\frac{1}{2} makes it easier to solve the algebraic equation for nn.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, while the sum of the exterior angles is always 360∘360^\circ.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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