2. A sector of a circle is the region bounded by two radii of the
circle and their intercepted arc.
In the figure, sector AOB is bounded by radii ππ΄
and ππ΅ and arc π΄π΅.
3. Theorem 1.
The ratio of the area A of a sector of a circle to the area of a
circle is equal to the ratio of the measures of the intercepted arc to
360.
In the figure,
π΄
ππ2 =
ππ΄π΅
360
or A =
ππ΄π΅
360
. ππ2
Theorem 2. Area of a Sector Theorem
If a sector has radius r and its arc has measure x, then the
are A is A =
π₯
360
. π . π2
4. Illustrative Examples:
A. Find the area of the sector as shown in the
figure.
Solution:
1. Sector KLM intercepts an arc whose
measure is 120Β°. The radius is 9.
A =
π₯
360
. π . π2
=
120
360
. (3.1416) . (9)2
A = 84.82
The area of the sector 84.82
5. 2. 1. Sector CPD intercepts an arc whose
measure is 80Β°. The radius is 4 ft.
A =
π₯
360
. π . π2
=
80Β°
360
(3.1416) (4 ππ‘)2
=
80Β°
360
(3.1416) (16ππ‘2)
A = 11.17 ft2
The area of a sector is 11.17 ft2.
6. Find the area of the sector as shown in the figure.
A =
240
360
. 3.1416 . 4ππ 2
A= 33.51 cm2
7. Find the area of the sector as shown in the figure.
1. A =
π₯
360
. π . π2
=
120
360
(3.1416) (4ππ)2
=
120
360
(3.1416) (16ππ2)
A = 16.76 cm2
8. A segment of a circle is the set of all points in the region
bounded by an arc of the circle and the chord of the arc.
In the figure, the shaded region is a segment of the
circle.
To find the area of a segment of a circle,
Area of segment = area of sector β area of triangle.
9.
10. Example 1.
Find the area of the segment of the circle in the given
figure.
Step1:
Find the area of the sector.
=
= 18.85 cm2
11. Step 2:
Find the area of the triangle.
A = Β½(bh)
Atriangle =
1
2
(3 3 cm x 6cm)
=
1
2
(3)(6) . 3
= 9 3 cm2
Asegment = 18.85 cm2 β 15.59 cm2
= 3. 26 cm2