Trigonometry

2. L E N G T H

3. A R E A

4. S E C T O R

5. ARC LENGTH AND
AREA OF A SECTOR

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;
• Recognize the formulas and steps for calculating the
arc length and area of a sector; and
• Apply the arc length and area of a sector on solving
problems in real life situation.

7. DEFINITION:
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 = 10
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. Calculate the radius and circumference of the
Earth.

11. Arc Length
Given:
𝑠 = 500 𝑚𝑖𝑙𝑒𝑠
𝜃 = 7.2° or
𝜋
25
𝑟 =?
Solution:
𝑠 = 𝑟𝜃
500 = 𝑟
𝜋
25
500
𝜋
25
=
𝑟
𝜋
25
𝜋
25
𝑟 =
12,500
𝜋
𝑚𝑖𝑙𝑒𝑠
𝑟 ≈ 3979 𝑚𝑖𝑙𝑒𝑠

12. Arc Length
Solution:
𝐶 = 2𝜋𝑟
𝐶 = 2𝜋 3979
𝐶 = 7958𝜋 𝑚𝑖𝑙𝑒𝑠
𝐶 ≈ 25,001 𝑚𝑖𝑙𝑒𝑠
Therefore, the approximate value of the circumference
of the Earth is 25,001 miles.

13. 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

14. 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
sq ft. or 1060.28 sq ft.
Therefore, the area of the lawn that receives water is
675𝜋
2
sq ft. or 1060.28 sq ft.

15. 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.

16. Arc Length & Area of a Sector
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.

17. Arc Length & Area of a Sector
Group 5: Find the area of a sector of a circle of radius 4 in., bounded by
two perpendicular arms. Verify the result using the formula for the area of
a circle.
Group 6: A voltmeter’s pointer is 6 centimeters in length. Find the angle
through which it rotates when it moves 2.5 centimeters on the scale.
Group 7: An umbrella has equally spaced 8 ribs. If viewed as a flat circle of
radius 7 units, what would be the area between two consecutive ribs of
the umbrella? (Hint: The area between two consecutive ribs would form a
sector of a circle)
Group 8: A circle with a diameter of 2 units is divided into 10 equal
sectors. Can you find the area of each sector of the circle?

18. Quiz
1. If a pizza pie has a diameter of 4 ft., what is the
length of the circumference?
2. A circle has a radius 9.5 cm. Find the length of
the arc intercepted by a central angle with
measure 120˚.
3. If the area of a sector of a circle is 8π and the
central angle is 45˚, find the radius of the circle.

19. Assignment
1. 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

20. 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.