Yes, as long as you know the angle of the arc (between the two radii at its ends at the centre of the arc):
arc_length = radius x angle_of_arc_in_radians
→ radius = arc_length ÷ angle_of_arc_in_radians
To convert between degrees and radians:
radians = π x degrees ÷ 180°
→ degrees = 180° x radians ÷ π
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You can use the cosine rule and the three lengths of the triangle to find the central angle X, (in radians). Then the length of the arc is r*X units.
To find the radius of a circle from a central angle of 120 degrees, you need additional information, such as the length of the arc or the area of the sector. If you have the arc length (s), you can use the formula ( r = \frac{s}{\theta} ), where ( \theta ) is in radians (120 degrees is ( \frac{2\pi}{3} ) radians). If you know the area of the sector, you can use ( r = \sqrt{\frac{A}{\frac{1}{2} \theta}} ), where ( A ) is the area and ( \theta ) is in radians. Without extra data, the radius cannot be determined solely from the angle.
Use the formula for the area of a circular sector, and solve for the angle.For a circular sector: area = radius squared times angle / 2 (Note: The angle is supposed to be expressed in radians; and in this specific problem, there is no need to convert it to degrees.) Since you know the area and the radius (according to the comments added to this question), you can solve for the angle. Once you know the angle (in radians!), the arc length is simply angle x radius.
The arcs are 59.31 cm and 3.49 cm.
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