Day 3 – Radians, Arc Length, and Sector Area
Radian: 1 radian is the angle of a sector when the radius of the circle is equal to its arc
length.
Example:
There is a relationship between the arc
length, the radius, and the sector angle.
radius
Let 𝜃 be equal to one radian; by definition
this is the angle that is subtended by an arc
length equal to the radius.
arc length
How many sectors of angle one radian can
be drawn in the circle?
We know: 𝐶 = 2𝜋𝑟
𝜋 radians =
We can use the relationship:
degrees
Example: Convert the following angles from degrees to radians:
a) 40
o
b) -100
o
Example: Convert the following angles from radians to degrees:
a)
4

3
b) 2.5
Arc Length and Sector Area:
We can use proportions to create identities relating arc length, sector area, and radius.
Example: Find arc length if:
o
a)   20 and r = 5 cm
b)
  2.1 and r = 10 cm
Example: Find the sector angle in a) radians and b) degrees:
r = 4 cm
a = 6 cm
Example: Find the sector area for the example above.
Challenge:
Angles in Standard Position: Angles rotating around the coordinate plane
Angles rotating in the counter-clockwise direction are
positive
Angles rotating in the clockwise direction are
negative
terminal arm
initial arm
Coterminal Angles:
 Angles having the same terminal arm
 Coterminal angles can be calculated by adding
o
or subtracting multiples of 360 or 2  radians
to known angles
Example: Find one positive and one negative coterminal angle for each of the following:
a) 120
o
b)
2
3
Principle Angle: the smallest, positive coterminal angle
Example: Find the principle angle for 800
o