Chords AB and AC contract the arcs at 60 and 120 degrees. The radius of the circle is R.

Chords AB and AC contract the arcs at 60 and 120 degrees. The radius of the circle is R. Find the area of the shaded shape.

Let us define the degree measure of the BC arc.

BC = 360 – 120 – 180 = 180, then BC is the diameter of the circle. BC = 2 * R see.

In the ABC triangle, the angle A = 90, since it is based on the diameter of the circle, the angle B = 120/2 = 60, the angle C = 60/2 = 30, since they are equal to half the degree measures of the arcs on which they rest.

Leg AB lies opposite angle 30, which means it is equal to half of CB. AB = CB / 2 = R.

Then AC ^ 2 = BC ^ 2 – B ^ 2 = (2 * R) ^ 2 – R ^ 2 = 3 * R ^ 2.

AC = R * √3 cm.

Determine the area of ​​the triangle ABC.

Savs = AB * AC / 2 = R * R * √3 / 2 = R2 * √3 / 2 cm2.

The area of ​​the BC sector is equal to half the area of ​​the circle.

Svs = Samb / 2 = (n * R ^ 2) / 2 =

Determine the area of ​​the figure.

S = Savs + Svs = R ^ 2 * √3 / 2 + n * R ^ 2/2 = R ^ 2/2 * (√3 + n) cm2.

Answer: The area of ​​the figure is R ^ 2/2 * (√3 + n) cm2.