Pre Calculus
Notes 6.1
Some Important Circle Measures
Definition of a Radian: A radian is the measure of an angle θ whose
arc length is equal to the radius of the circle.
r
r
θ
r
Because the circumference of the unit circle is 2π, we have the equation
2π radians = 360°, or
π radians = 180°
Example 1:
Express in radian measure:
Express in degree measure:
(a) 200°
(d)
radians
(b) -72°
(e)
radian
(c) 45°
(f) 1 radian
1
The length of an arc intercepted by a central angle of θ radians in a circle
of radius r is rθ.
s
θ
r
Ex 2: Find the length of an arc that subtends a central angle of 45° in a circle of
radius 10 m.
Ex 3: Memphis, Tennessee and New Orleans, Louisiana lie approximately on the
same meridian. Memphis has latitude 35°N and New Orleans 30°N. Find the
distance between these two cities if the radius of the earth is 3960 mi.
2
Area of the Sector of a Circle
In a circle of radius r, the area of a sector with central angle radians is
Ex 4: A sector of a circle has a central angle of 60°. Find the area of the sector
if the radius of the circle is 3 mi.
Ex 5: A sector of a circle of radius 24 m has an area of 288 m². Find the central
angle of this sector.
Ex 6: Three circles with radii 1, 2, and 3 ft are externally tangent to one another,
as shown in the figure. Find the area of the sector of the circle of radius 1
that is cut off by the line segments joining the center of that circle to the
centers of the other two circles.
2
3
1
1
3