6. OBJECTIVES:
• State and illustrate the theorem on length of an arc
as well as other related formulas for central angle θ
and radius of the circle; and
• Accurately solve problems involving arc length and
area of a sector.
7. An arc is a portion of the
circumference of a circle.
Arc length is defined as the
length of an arc, s, along a
circle of radius r subtended
(drawn out) by an angle.
Arc Length
x
y
8. THEOREM: On a circle of radius r, a central angle (an
angle whose vertex is the center of the circle) of θ
radians intercepts an arc, whose length is equal to the
product of θ and r.
Arc Length Formula:
𝑠 = 𝑟𝜃
Arc Length
9. Arc Length
What is the length of the arc that measures 60° in a circle of radius
10cm?
Given:
𝜃 = 60° or
𝜋
3
, 𝑟 = 10𝑐𝑚, 𝑠 =?
Solution:
𝑠 = 𝑟𝜃
𝑠 = 10𝑐𝑚
𝜋
3
𝑠 =
10𝜋
3
𝑐𝑚
Therefore, the arc length is
10𝜋
3
cm or 10.47 cm.
10. Arc Length
Eratosthenes of Cyrene (276-194 BC) was a Greek
scholar who lived and worked in Cyrene in Alexandria.
One day while visiting in Syene he noticed that the
sun’s rays shone directly down a well. On this date 1
year later, in Alexandria, which is 500 miles due north
of Syene he measured the angle of the sun about 7.2
degrees. Given the information, calculate the radius
and circumference of the Earth.
12. Arc Length
Solution:
𝐶 = 2𝜋𝑟
𝐶 = 2𝜋 3979 𝑚𝑖𝑙𝑒𝑠
𝐶 = 7958𝜋 𝑚𝑖𝑙𝑒𝑠
𝐶 ≈ 25,001 𝑚𝑖𝑙𝑒𝑠
Therefore, the approximate value of the circumference
of the Earth is 25,001 miles.
13. Arc Length
Group 1: The length of the pendulum is 40 in. and swings through an angle of 20˚.
Through what distance does the bob swing from one extreme point to another?
Group 2: An arc with a degree measure 60˚, has an arc length of 5π cm. What is
the radius of the circle on which the arc sits?
Group 3: Lisa orders a slice of pizza. Its central angle is 68˚. The distance from the
vertex of the pizza to the edge of its crust is 18 cm. Find the approximate length of
the crust (arc) of the pizza.
Group 4: Find the distance that the earth travels in one month in its path around
the sun. Assume the path of the earth around the sun is a circle of radius 93
million miles.
Group 5: If a pizza pie has a diameter of 4 ft., what is the length of the
circumference?
14. The area of a sector of a
circle is another geometric
application of radian. A
sector is a part of a circle
between two radii with the
given central angle.
𝐴 =
1
2
𝑟2𝜃
Area of a Sector
x
y
15. Area of a Sector
A water sprinkler sprays water over a distance of 30 ft. while rotating the through
an angle of 135˚. What area of lawn receives water?
Given:
r=30 ft., θ=135˚ or
3𝜋
4
, A=?
Solution:
𝐴 =
1
2
𝑟2
𝜃
𝐴 =
1
2
(30)2 3𝜋
4
𝐴 = 450
3𝜋
4
𝐴 =
675𝜋
2
ft. or 1060.28 ft.
Therefore, the area of the lawn that receives water is
675𝜋
2
ft. or 1060.28 ft.
16. Area of a Sector
Crops are often grown using a technique called center pivot irrigation that results in
circular shaped fields. If the irrigation pipe is 450m in length, what is the area that can
be irrigated after a rotation of 240 degrees?
Given:
r=450 m, θ=240˚ or
4𝜋
3
, A=?
Solution:
𝐴 =
1
2
𝑟2𝜃
𝐴 =
1
2
(450)2 4𝜋
3
𝐴 = 101,250
4𝜋
3
𝐴 = 135000𝜋 𝑚2 or 424, 115 𝑚2
Therefore, the area that can be irrigated after a rotation of 240 degrees is 424, 115 𝑚2.
17. Quiz
Problem: For a 60-foot Little League Baseball field, the distance from home
base to the nearest fence (or other obstruction) on fair territory should be a
minimum of 200 feet. The commissioner of parks and recreation is making
plans for a new 60-foot field. Because of limited ground availability, he will use
the minimum required distance to the outfield fence. To increase safety,
however, he plans to include a 10-foot wide warning track on the inside of the
fence. To further increase safety, the fence and warning track will extend both
directions into foul territory. In total the arc formed by the outfield fence
(including the extensions into the foul territories) will be subtended by a central
angle at home plate measuring 96˚.
(a). Determine the length of the outfield fence.
(b). Determine the area of the warning track.
18. Assignment
Given the concentric circles
with center A and with 𝑚∠𝐴 =
60˚, calculate the arc length
intercepted by ∠𝐴 on each
circle. The inner circle has a
radius of 10, and each circle
has a radius 10 units greater
than the previous circle.
x
y
19. Reference:
Beltran, O. M., Ymas, S. E., & Dela Cruz, V. O. (2007).
Plane Trigonometry. pp. 173-176.
https://mathbooks.unl.edu/PreCalculus/arc-
length.html#:~:text=Arc%20length%20is%20defined%2
0as,out)%20by%20an%20angle%20%CE%B8.
