3 Triangles in a cirle..

Geometry Level 2

In the adjoining figure O is the centre of the circle with radius 4. AB,CD and EF are the diameters of the circle. Angle OAF= angle OCB=60.

What is the area of the shaded region ? (Correct to two decimal places )


The answer is 4.34.

Tanmay Goyal by tanmay goyal , 21, India

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2 solutions

The shaded portion is composed of three congruent segment of a circle. The area of a segment of a circle is equal to the area of a sector minus area of the triangle. So we have

A s h a d e d = 60 360 ( π ) ( 4 2 ) − 1 2 ( 4 2 ) ( sin ⁡ 60 ) ( 3 ) ≈ 4.34813 A_{shaded}=\dfrac{60}{360}(\pi)(4^2)-\dfrac{1}{2}(4^2)(\sin 60)(3)\approx 4.34813

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Blink Eighteen
Feb 23, 2015

The triangles are equilateral. The area of each triangle (sqrt(3)/4) r r

Area of the Circular Segment

= (1/6)Area of circle - area of the triangle

= (1/6) Pi r r - (sqrt(3)/4) r*r

= [Pi/6 - sqrt(3)/4 ] * 4 * 4 = 1.4493768

There are 3 such areas so final answer = 4.3481

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