Maths Olympiad Prep

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Geometry Difficulty 4.0 AIME Find the answer United States

Problem:
A polygon is regular if all its sides and angles are the same. Find the measure of each angle in a regular dodecagon (12-sided polygon).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Figure 1
A regular dodecagon can be divided into 1010 triangles, as shown above. The sum of the angles in each triangle is 180∘180^{\circ}, so the sum of the angles in the dodecagon is 10⋅180∘=1800∘10 \cdot 180^{\circ} = 1800^{\circ}. Since it is regular, all the angles are the same, so we divide by 1212 to get 1800∘/12=150∘1800^{\circ} / 12 = 150^{\circ} for each angle.

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