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Difficulty: MediumPolygon Angles and Properties

The measures of the six interior angles of a convex hexagon are in the ratio 3:4:5:5:6:73:4:5:5:6:7. What is the measure, in degrees, of the largest interior angle of this hexagon?

  1. A
    7272
  2. B
    8484
  3. C
    126126
  4. D
    144144
  5. 168168Answer

Answer

168
The sum of the interior angles of any convex nn-gon is given by (n−2)×180∘(n-2) \times 180^\circ. For a hexagon (n=6n=6), this sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. The angles are in the ratio 3:4:5:5:6:73:4:5:5:6:7, which sum to 3+4+5+5+6+7=303+4+5+5+6+7 = 30 parts. Each part corresponds to 720∘÷30=24∘720^\circ \div 30 = 24^\circ. The largest angle is represented by the largest part of the ratio, 7, which equals 7×24∘=168∘7 \times 24^\circ = 168^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
720∘720^\circ
The sum of the interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For a hexagon (n=6n=6), the sum is (6−2)×180∘=4×180∘=720∘(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Determine the total number of parts in the given ratio.
30 parts
Sum the components of the ratio: 3+4+5+5+6+7=303 + 4 + 5 + 5 + 6 + 7 = 30.
3
Find the value of one part in degrees.
24∘24^\circ per part
Divide the total interior angle sum by the total number of ratio parts: 720∘÷30=24∘720^\circ \div 30 = 24^\circ.
4
Calculate the measure of the largest interior angle.
168∘168^\circ
The largest angle corresponds to the largest component in the ratio, which is 7. Multiply the value of one part by 7: 7×24∘=168∘7 \times 24^\circ = 168^\circ.

Key Concept

Calculating the interior angle measures of an irregular convex polygon using the polygon interior angle sum formula and a given ratio.
Estimated Time:1m 30s
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Topics

Polygon Angles and Properties

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