The chord of the circle = 6 and subtracts 120 degrees. Find the arc length, sector area?

Let’s draw circles OA and OB.

The central angle AOB is equal to the value of the arc AB on which it rests.

Angle AOB = 120.

The AOB triangle is isosceles, since ОА = ОВ = R see.

Let us apply the cosine theorem for the triangle AOB.

AB ^ 2 = R ^ 2 + R ^ 2 – 2 * R * R * Cos120.

36 = 2 * R ^ 2 – 2 * R ^ 2 * (-1/2) = 3 * R ^ 2.

R ^ 2 = 12 cm.

R = 2 * √3 cm.

Let us determine the length of the smaller arc AB.

Lav = n * R * 120/180 = n * 2 * √3 * 120/180 = 4 * n * √3 / 3.

Let us determine the area of the OAB sector.

Ssec = n * R2 * 120/360 = n * (2 * √3) 2 * 120/360 = 4 * n cm2.

Answer: The length of the arc is 4 * n * √3 / 3, the area of the sector is 4 * n cm2.