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therefore its arc length must be 3 times which we can write as . Similarly, we could solve for the radius if we knew the arc length and the radian measure.
That relationship only holds because we are using radian angle measure. So how could we compute the arc length if we measured our angle instead in degrees? One way would be to rst compute the circumference of the full circle with that radius, . The full circumference would be . Then we can multiply that full circumference by what fraction of the circle we have, which is the degree measure of our angle divided by . For example, if our arc had a radius of 3 and an angle measure of , the arc length would be times . That
would be the same as . This is the same arc length as our rst example which must mean that a radian measure of is the same as a degree measure of .
Another useful strategy with problems involving angle measures in circles is to convert from degree measure to radian measure. The angle that traces out the whole circumference of a circle is radians because the circumference is times the radius. That means each 1 degree is radians divided into 360 parts. So 1 degree is radians. So if we are given the
degree measure, we can always nd the radian measure by multiplying by radians per degree. For example, 45 degrees times radians per degree is equal to radians, which is the same radians we saw before. If we want to know the degree measure from a given radian measure, we can think through it the other way. Every radians is 360 degrees, so every 1 radian is degrees. To get the degree measure, we multiply the radian measure by
degrees per radian. For example, 2 radians times degrees per radian is equal to degrees, which is about 114.6 degrees.
Unit 7
Lesson 3: Measure That Arc
35