Length of a Circular Arc

An angle whose radian measure is θ is subtended by an arc that is the fraction θ/(2π) of the circumference of a circle. Thus, in a circle of radius r, the length s of an arc that subtends the angle θ (see the Figure on the right) is

θ θ s = × circumference of circle = (2πr) = θr 2π 2π

Solving for θ, we get the important formula s θ = r

5 The formula above allows us to deﬁne radian measure using a circle of any radius r: The radian measure of an angle θ is s/r, where s is the length of the circular arc that subtends θ in a circle of radius r (see the Figures below).

EXAMPLE: (a) Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30◦. (b) A central angle θ in a circle of radius 4 m is subtended by an arc of length 6 m. Find the measure of θ in radians. Solution: π (a) We know that 30◦ = . So the length of the arc is 6 π 5π s = rθ = (10) = m 6 3

(b) By the formula θ = s/r, we have s 6 3 θ = = = rad r 4 2

EXAMPLE: (a) Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15◦. (b) A central angle θ in a circle of radius 9 m is subtended by an arc of length 12 m. Find the measure of θ in radians. Solution: π (a) We know that 15◦ = . So the length of the arc is 12 π 7π s = rθ = (21) = m 12 4

(b) By the formula θ = s/r, we have s 12 4 θ = = = rad r 9 3

6 Area of a Circular Sector

The area of a circle of radius r is A = πr2. A sector of this circle with central angle θ has an area that is the fraction θ/(2π) of the area of the entire circle (see the Figure on the right). So the area of this sector is

θ θ 1 A = × area of circle = (πr2) = r2θ 2π 2π 2

EXAMPLE: Find the area of a sector of a circle with central angle 60◦ if the radius of the circle is 3 m. Solution: To use the formula for the area of a circular sector, we must ﬁnd the central angle of the sector in radians: ( ) π π 60◦ = 60 rad = rad 180 3 Thus, the area of the sector is ( ) 1 1 π 3π A = r2θ = (3)2 = m2 2 2 3 2

EXAMPLE: Find the area of a sector of a circle with central angle 4◦ if the radius of the circle is 45 m. Solution: To use the formula for the area of a circular sector, we must ﬁnd the central angle of the sector in radians: ( ) π π 4◦ = 4 rad = rad 180 45 Thus, the area of the sector is ( ) 1 1 π 45π A = r2θ = (45)2 = m2 2 2 45 2

EXAMPLE: A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of 120◦ (see the Figure on the right). Find the area of the fairway watered by the sprinkler.

Solution: First convert 120◦ to radian measure as follows. ( ) π 2π θ = 120◦ = 120 rad = rad 180 3 Therefore ( ) 1 1 2π 4900π A = r2θ = (70)2 = ≈ 5131 ft2 2 2 3 3