The document provides instruction on calculating arc length, circumference, sector area, and tangent lines of circles. It defines key terms like arc, radius, diameter, chord, sector, and tangent. It presents the formulas for finding arc length, circumference, and sector area. Examples are worked through applying the formulas to find values like arc length, sector area, and unknown angles. Pythagoras' theorem is used in one example to find the length of a tangent line. Key circle properties discussed include a tangent line being perpendicular to the radius and a semi-circle angle always being 90 degrees.

1. Int 2
Working with Sectors
1

2. Arc length of a circle
Int 2
Aim of Today’s Lesson
To find and be able to use the formula
for calculating the length of an arc.
Monday 28 January 2013 2

3. Arc length of a circle
Int 2
Q. What is an arc ?
Answer
A
An arc is a fraction
of the circumference.
minor arc
B
major arc
Monday 28 January 2013 3

4. Arc length of a circle
Int 2
Q. Find the circumference of the circle ?
Solution
10cm
C = πD
C = 3.14 × 20
C = 62.8cm
Monday 28 January 2013 4

5. Arc length of a circle
Int 2
Q. Find the length of the minor arc XY below ?
x Arc length Arc angle
connection =
πD 360o
y
o 6 cm
45
45o
arc length = o
× (3.14 × 12)
360
o 360
arc length = 4.71cm
Monday 28 January 2013 5

6. Arc length of a circle
Int 2
Q. Find the length of the minor arc AB below ?
Arc length Arc angle
=
A connection πD 360o
9 cm
60o
arc length = × (3.14 × 18)
o
60
o
360
B arc length = 9.42cm
Monday 28 January 2013 6

7. Arc length of a circle
Int 2
Q. Find the length of the major arc PQ below ?
Arc length Arc angle
=
A connection πD 360o
10 m
260
o
260o
arc length =
o
100
o
× (3.14 × 20)
360
B arc length = 45.38cm
Monday 28 January 2013 7

8. Sector area of a circle
Int 2
Aim of Today’s Lesson
To find and be able to use the formula
for calculating the sector area of a circle.
8

9. Area of Sector in a circle
Int 2
A
minor sector
B
major sector
9

10. Area of Sector in a circle
Int 2
Q. Find the area of the circle ?
Solution
10cm
A = πr 2
A = 3.14 × 102
A = 314cm 2
Monday 28 January 2013 10

11. Area of Sector in a circle
Int 2
Find the area of the minor sector XY below ?
x connection
Area Sector = Sector angle
y πr2 360o
o 6 cm
45
45o
Area of Sector = × (3.14 × 62 )
360
o
360o
Area Sector = 14.14cm 2
Monday 28 January 2013 11

12. Area of Sector in a circle
Int 2
Q. Find the area of the minor sector AB below ?
connection
A Area Sector Sector angle
=
πr2 360o
9 cm
60o
Area Sector = × (3.14 × 9 )
2
o
60
o
360
B
Area Sector = 42.41cm 2
Monday 28 January 2013 12

13. Area of Sector in a circle
Int 2
Q. Find the area of the major sector PQ below ?
connection
Sector Area Sector angle
=
A πr2
360o
10 m
260
o 260o
Sector Area = × (3.14 × 102 )
o
100
360o
B
Area Sector = 226.89cm 2
Monday 28 January 2013 13

14. Arc length of a circle
Int 2
Q. The arc length is 7.07cm. Find the angle x o
Arc length Arc angle
=
A connection πD 360o
9 cm
x
o 7.07
x =
o
× 360o
(3.14 × 18)
B
x o = 45o
Monday 28 January 2013 14

15. Area of Sector in a circle
Int 2
Q. Find the angle given area of sector AB is 235.62cm2 ?
connection
Sector Area Sector angle
=
A πr2
360o
10 m
x
o
235.62
x =
o
× 360o
(3.14 × 102 )
B
xo = 270o
Monday 28 January 2013 15

16. Extension - Thinking
Int 2
Monday 28 January 2013 Created by Mr Lafferty
16

17. Tangent line
Int 2
The radius of the circle that touches the tangent
line is called the point of contact radius.
Online Demo
Special Property
The point of contact radius
is always perpendicular
(right-angled)
to the tangent line.
Monday 28 January 2013 17

18. Tangent line
Int 2
Q. Find the length of the tangent line between
A and B.
Solution
Right-angled at A since B
AC is the radius at the point
of contact with the 10
Tangent.
By Pythagoras Theorem we have
A C
a 2 + b2 = c 2 8
a + 8 = 10
2 2 2
a 2 = 102 − 82
a 2 = 100 − 64 = 36
a = 36 = 6
Monday 28 January 2013 18

19. Summary of Circle Topic
Arc length is
Int 2 Pythagoras Theorem
centre angleo
Arclength = × πD line that bisects a chord SOHCAHTOA
360o
1. Splits the chord into 2
equal halves.
2. Makes right-angle with
the chord.
3. Passes through centre
of the circle
Circumference Semi-circle angle is
is
C = πD
o
always 90
Area is
Tangent touches
circle at one A = πr 2
point
and make angle
o
90 with point of
contact radius
Diameter Radius
1
D = 2r r = D
2 Sector area
centre angleo
Sector = o
× πr 2
Monday 28 January 2013 360 19