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 45º
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)

6.3

The difference between an arc's length and measure lies in their definitions and what they represent.

The "measure" of an arc refers to the angle that it subtends from the center of the circle. We can denote this angle as A. It is essentially the measure of the central angle that corresponds to the arc.

On the other hand, the "length" of an arc represents the actual distance along the circumference of the circle that the arc covers. It is the curved segment of the circle between the two endpoints of the arc. The length of an arc depends on the radius of the circle and the measure of the arc.

To find the measure of an arc, you need to know the central angle that the arc represents. This angle can be measured in radians or degrees. If the angle is measured in radians, the measure of the arc is given by the formula R * A, where R is the radius of the circle and A is the measure of the angle in radians. If the angle is measured in degrees, the formula becomes (pi * R * A) / 180, where A is the measure of the angle in degrees.

To find the length of an arc, you need to know the radius of the circle and the measure of the angle subtended by the arc. The length is given by the formula 2 * pi * R * (A / 360), where R is the radius of the circle and A is the measure of the angle in degrees. In the case where the angle is measured in radians, the formula becomes R * A, as there are 2 * pi radians in a full circle.

For example, let's say you have a pizza that has been cut into 8 equal pieces. If you look at one of those pieces, the arc measure (angle) would be 45 degrees, as each piece represents 1/8th of a circle. The arc length (length of the curved part of your pizza slice) would be 1/8th of the circumference of the pizza, which is equal to (2 * pi * r) / 8, where r is the radius of the pizza.