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Whats the difference between an arc [of a circle]'s length and measure? And how do you find each of them?

The "measure" of an arc is the angle that it subtends from the center of the circle. Call that angle A.

The length of the arc is
2 pi R * [A/(2 pi)] = R*A, if A is in radians and R is the radius of the circle, or pi R*A/180, if A is measured in degrees.

Emily think of a pizza that has been cut into 8 equal pieces, and look at one of those pieces.

The arc measure would be the angle that your piece of pizza has as its point, in this case 45ยบ
The arc length would be the length of the curved part of your pizza slice, in this case 1/8 the circumference of the pizza
(remember C=2*pi*r)

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