Help with finding arc lengths?

Question

when you find arc lengths, do you use the inscribed angle measurement, the central angle measurement, or does using either one work fine?

Answer 1

The length of an arc is normally defined by the central angle.

On Earth, the "arc length" on the surface that corresponds to a central angle of one minute (1/60 degree) is called the "nautical mile"

In the "grade" system (400 grades to a full circle), a central angle of 1/100 of a grade (a centigrade) corresponds to 1 km on the Earth surface.

The Roman mile (a.k.a. the statute mile) began as a measure of 1000 steps (left foot to left foot) by a marching army. Its modern measure includes a slight adjustment made after the Middle Ages, when it was thought that 60 miles was very close to one degree in latitude. Turns out to be 15% too short.

To find the length of an arc from the central angle x in degrees, knowing the radius r:
length = r * pi * x / 180

If the angle x is given in radians: length = x * r

It is also possible to find the length from the inscribed angle, keeping in mind that the inscribed angle will be either x/2 or 180 - x/2, depending on which side of the chord it is inscribed.

Answer 2

The easiest way is to use the central angle. For a radius r and a central angle θ, the formula for arc length is rθ. As a check, note that the central angle for the whole circle would be 2π, so the arc length for the whole circle would be 2πr.

If you use the inscribed angle α, the arc subtended by that angle would be 2α. So the formula for arc length would be 2rα.

One final note. The formulas above require the angles to be measured in radians.

Answer 3

Circumference = 2 x pi x r Arc forms an angle at the center. Circumference forms an angle of 360* at the center. Angle formed by arc / 360 = length of the arc / circumference Length of arc is proportionate to the angle formed at the center

Answer 4

one is just as good as the other