Final answer:
The length of arc AB can be found using the formula As = r × A°, with angle A° in radians and r as the radius. For small arcs, arc length is approximately equal to the straight-line distance. Accurate calculations require using the formula for longer arcs.
Step-by-step explanation:
To find the length of arc AB on a circle with the center at the origin, we need to know the radius (r) of the circle and the angle in radians (A°) that the arc subtends at the center. The arc length (As) is calculated by the formula As = r × A°, where the angle must be in radians for the formula to work. If the angle is given in degrees, it must first be converted to radians by multiplying by π/180.
If the arc spans a small angle, it can be approximated that the arc length is nearly equal to the straight-line distance between two points on the circumference. This approximation is represented by the equation c ≈ c', where c is the chord length between two points and c' is the arc length. However, for precise calculations, especially when the arc covers a significant portion of the circle, the approximation should not be used, and the arc length should be calculated using the aforementioned formula.
It is important to note that the arc length represents the distance traveled along the circular path and not the straight-line distance between two points unless the arc is very small.