In a circle of radius R, two parallel chords are drawn on one side of the center, one of which contracts the arc at 120 °

| May 8, 2021 | education

In a circle of radius R, two parallel chords are drawn on one side of the center, one of which contracts the arc at 120 °, and the other at 60 °. Determine the part of the area of the circle enclosed between the chords.

Since the chord СD contracts the arc 120, the central angle COD = 120.

Since the chord KM contracts the arc 60, the central angle KOM = 60.

Let us determine the area of ​​the OSKMDO sector S1 = n * R ^ 2 * 120/360 = n * R ^ 2/3 cm2.

Determine the area of ​​the OMEC segment S2 = n * R ^ 2 * 60/360 = n * R ^ 2/6 cm2.

Determine the area of ​​the triangle СОD Sсod = (R ^ 2 * Sin120) / 2 = R ^ 2 * √3 / 4 cm2.

Determine the area of ​​the triangle KOM Scom = (R ^ 2 * Sin60) / 2 = R ^ 2 * √3 / 4 cm2.

Determine the area of ​​the KEM segment. Skem = S ^ 2 – Scom = (n * R ^ 2/6) – (R ^ 2 * √3 / 4) cm2.

Let’s define the area between the chords CD and KM. Sscmd = S1 – Ssod – Skem = (n * R ^ 2/3) – (R ^ 2 * √3 / 4) – (n * R ^ 2/6) + (R ^ 2 * √3 / 4) = (n * R ^ 2/3) – (n * R ^ 2/6) = n * R ^ 2/6 cm2.

Answer: The area enclosed between the chords is equal to n * R ^ 2/6 cm2.



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