There Are 3 Interior Angles! Which One?

Geometry Level 1

The above shows an isosceles triangle with its apex at the center of a circle and its base tangent to the circle. If one of the isosceles angles of the triangle is 3 0 ∘ 30^\circ , then the area of the overlapping region bounded by the triangle and the circle is __________ \text{\_\_\_\_\_\_\_\_\_\_} times the area of the circle.

1 2 \frac12 1 3 \frac13 1 4 \frac14 1 5 \frac15

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Relevant wiki: Circles Problem Solving - Basic

Short cut for MCQ:

If the unequal angle is 30 degrees, ratio of area of overlapping to non-overlapping is 30 360 = 1 12 \dfrac{30}{360} = \dfrac{1}{12} . There is no option of 1 12 \dfrac 1 {12} .

Therefore the equal angles are 30 degrees and non-equal angles is 120 degrees.

Therefore required ratio = 120 360 = 1 3 =\dfrac{120}{360}=\dfrac{1}{3} .

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