Week 9 – SS1 Second Term Mathematics Notes

WEEK NINE
TOPIC

Introduction of circle and its properties
Calculation of length of arc and perimeter of a sector
Area of sectors and segments. Area of triangles

(a) Introduction of circle and its properties
Parts of a circle: The figure below shows a circle and its parts.

The centre is the point at the middle of a circle. The circumference is the curved outer boundary of the circle. An arc is a curved part of the circumference. A radius is any straight line joining the centre to the circumference. The plural of radius is radii. A chord is any straight line joining two points on the circumference. A diameter is a straight line which divides the circle into two equal parts or a diameter is any chord which goes through the centre of the circle.
Region of a circle
The figure below shows a circle and its different regions.
A sector is the region between two radii and the circumference. A semi-circle is a region between a diameter and the
Semi – circle

Segment
circumference i.e half of the circle. A segment is the region between a chord and the circumference.

EVALUATION
Draw a circle and show the following parts on it. Two radii, a sector, a chord, a segment, a diameter, an arc; label each part and shade any regions.

(b) Calculation of length of arc and perimeter of a sector
Given a circle centre O with radius r. The circumference of the circle is 2Пr. Therefore, in the figure below, the length, L, of arc XY is given as:
L = θ x 2Пr
360^o

Where θ is the angle subtended at the centre by arc XY and r is the radius of the circle.

Also,
The perimeter of Sector XOY = r + r + L
Where

L = length of arc XY
= θ X 2 Пr
360

Then
Perimeter of
Sector XOY = r + r + L
= 2r + θ x 2 Пr
360^o

EXAMPLES

An arc of length 28cm subtends an angle of 24⁰ at the centre of a circle. In the same circle, what angle does an arc of length 35cm subtend?
Calculate the perimeter of a sector of a circle of radius 7cm, the angle of the sector being 108^o, if П is .

Solutions
1. L = θ x 2 Пr
360
When L = 28cm , θ = 24⁰, r = ?
Then

L = θ x 2 Пr
360^o

28 = 24 x 2 x 22 x r
360⁰ 7
Cross-multiply:
24 x 44 x r = 28 x 360 x 7

15
7 60
r = 28 x 360 x 7 cm
24×44
4 11

r = 49 x 15 cm
11
r = 735 cm
11
Also
When L = 35cm, r = 735 cm
11
θ = ?

Then
L = θ x 2 Пr
360^o

35 = θ x 2 x 22 x 735
360 7 11

Then,
Cross multiply

x 360 x 7 x 11 = θ x 44 x 735

1 11
35 x 360 x 77 = θ
44 x 735
4 105 3

360 = θ = 30⁰
12
Thus, when the length of the arc is 35cm, the angle subtended at the centre is 30⁰2. Perimeter of a sector of a circle = 2r + θ x 2 Пr
360^o

= 2 x7 + 108 x 2 x 22 x 7
         360 7 1
         3
= 14 + 108 x 44cm
    360 10
= 14 + 3 x 44 cm
10
= 14 + 132 cm
10
= 14 + 13.2 cm
= 27.2 cm

EVALUATION
1. A piece of wire 22cm long is sent into an arc of a circle of radius 4 cm. What angle does the wire subtend at the centre of the circle?
2. Calculate the perimeter of a sector of a circle of radius 3.5cm, the angle of the sector being 162⁰ if П is 22
7.

Length of chord and perimeter of a segment.
Consider a circle centre O with radius r

If OC is the perpendicular distance from O to chord AB and angle
AOB = 2 θ, then the length of chord AB can be found as follows:

In right-angled triangle OCA

AC = Sin θ
r
Cross multiply:
__
AC = r Sin θ
Since
AB = 2 x AC
AB = 2r Sin θ

Where
r = radius of the circle
θ =Semi Vertical angle of the sector i.e half of the angle subtended at the centre by arc AB.
Also
The perimeter of segment ACBD = Length of chord AB + length of arc ADB
= 2r Sin θ + θ x 2 Пr
360^o
Example
In a circle of radius 6 cm, a chord is drawn 3cm from the centre.
(a) Calculate the angle subtended by the chord at the centre of the circle.

(b) Find the length of the minor arc cut off by the chord

Hence find the perimeter of the minor segment formed by the chord and the minor arc.

Solution

a. Let the required angle
= AOB = 2 θ
Where
θ = Semi vertical angle of the sector.

Then
Cos θ = 3cm = 1
6cm 2

Cos θ = 0.5000
θ = Cos^-10.5000
θ = 60⁰-: Required angle = 2 θ
             = 2 x 60⁰
            = 120⁰
b Length of minor arc ADB =θ x 2 Пr
1         2     360⁰
= 120 x 2 x 22 x 6cm
360 7
3
1
= 4 x 22cm
7
= 88cm = 12 4cm
7 7

Perimeter of minor segment ACBD
= Length of + length of arc
Chord AB ADB
= 2r Sin θ + 12
= 2 x 6 x sin60⁰ + 12
= 12 x Sin 60⁰ + 12
= 12 x 0.8660 + 12.5714cm
= 10.3920 + 12.5714cm
= 10.3920
12.5714
22.9634cm
= 22.96 cm to 2 places of decimal.
EVALUATION
1. a. A chord 4.8cm long is drawn in a circle of radius 2.6cm. Calculate the distance of the chord from the centre of the circle.
  b. Calculate the angle subtended at the centre of the circle by the chord in Question 1(a) above
  c. Hence find the perimeter of the minor segment formed by the chord and the minor arc of the circle.
READING ASSIGNMENT
NGM SS BK 2, pg. 31,Ex2a, Nos.2,3,5.
(c) Area of sectors and segments. Area of triangles
Area of sectors
Area of a sector of a circle is given by the formular;
Area of sector θ x πr ²
360^o
where r = radius of the circle, θ = angle subtended at the centre by XY or angle of the sector

Examples
1. Calcualte the area of sector of a circle which subtends an angle of 45^o at the centre of the circle, diameter 28cm (π = 22/7).
2. The area of a circle PQR with centre O is 72cm². What is the area of sector POQ, if POQ = 40^o?
Solutions
1. Since the diameter of the circle = 28cm
  d = 2r = 28
  where d = diameter and r = radius
  thus 2r = 28
  2r = 28 = 14cm
  1. 2
Area of sector = θ x πr ²
360^o
= 45 x 22 x ( 14 ) ²
360 7
= 1/8 x 22/7 x 14 x 14 cm
= 77cm²2.

Since the area of the whole circle PQR = 72cm²
Then
Area of sector = θ x πr²
360^o
But πr²= Area of the whole circle PQR = 72cm²
:. Area of = 40 x 72cm²
sector POQ 360^o
    = 8cm²
Evaluation
complete the table below for areas of sectors of circles. make a rough sketch in each case.

Radius Angle of sector Area of sector
a.14cm – 462cm²
b. — 140 99cm²
Area of segments
A segments of a circle is the area bounded by a chord and an arc of the circle.Considering the figure below, we have a major segment and a minor segment .

Given the diagram below:
Area of the shaded segment= Area of sector POQ – Area of triangle POQ
= θ
360^o x πr² – ½ r² sin θ

Where
r = radius of the circle
θ = angle subtended by the sector at the centre
Π= a constant = 22/7

Examples
1. calculate the area of the shaded segment of the circle shown below:

2.Calculate the area of the shaded parts in the figure below. All dimensions are in cm and all arcs are circular.

Solutions
1 Area of the given shaded segment =θ x πr² – 1/2r² Sin θ
360^o
    = 56/360 x 22/7 x ( 15 ) ² – ½ x ( 15)² sin 56⁰
= 1/45 x 22 x 15 x 15 – ½ x 15 x 15 sin 56^o
= 22 x 5 – ½ x 225 x sin 56⁰
= 110 – ½ x 225 x 0.8290
= 110 – 225 x 0. 4145
= 110 – 93.2625cm²
= 110 – 93.2625cm²
= 16.7375cm²
= 16.7cm² to 3 s. f
2)

The arc in the given figure is part of a circle as shown in the figure above. Thus area of given shaded segment = Area of sector – area of triangle
     = θ x πr² – ½ r² sin θ
360
= 90/ 360 x 22/7 x ( 14) ² – ½ x (14) ² sin 90^o
=¼ x 22/7 x 14 x 14 – ½ x 14 x 14 x 1
= 11 x 14 – 14 x 7 cm²
= 154 – 98cm²
= 56cm²
EVALUATION
Calculate the area of the shaded parts in the figure below. All dimensions are in cm and all arcs are circular.
1. b)
GENERAL EVALUATION
1. An arc of a circle radius 7cm is 14cm long. What angle does the arc subtend at the centre of the circle?
2. An arc of a circle whose radius is 10cm subtends an angle 60⁰ at the centre. Find the length of the arc.
3. In the diagram below, O is the centre of the circle of radius 20cm. Calculate:
  1. The area of the minor segment PQ
  2. The area of the major segment PQ
  3. The perimeter of the minor segment. (take = 3.13)
READING ASSIGNMENT
NGM SS BK1 Pages 134-139 Ex 12d Nos 6 and 9 139

WEEKEND ASSIGNMENT
1. Calculate the area of a sector of a circle of radius 6cm which subtends an angle of 70^oat the centre (π = 22/7) A. 44cm² B. 22cm² C. 66cm² D. 11cm² E. 16.5cm²
2. What is the angle subtended at the centre of a sector of a circle of radius 2cm if the area of the sector is 2.2cm²? (π = 22/7)A. 120^o B. 31 ½^o C. 43^o D. 58^o E. 63^o
3. What is the radius of a sector of a circle which subtends 140^o at its centre and has an area of 99m²? A. 18m 27m C 9m E. 30m E. 24m
4. A sector of 80^o is removed from a circle of radius 12cm What area of the circle is left? A. 253cm² B. 704cm²C 176cm²D. 125cm²E. 352cm²π
5. Calculate the area of the shaded segment of the circle shown in the figure below:
  ( π = 22/7 )
A. 10.45cm² B. 20.90cm²C. 5.25cm²D. 19.0cm²E. 17.45cm²
THEORY
1. The figure below shows the cross section of a tunnel. It is in the shape of a major segment of a circle of radius 1m on a chord of length 1.6m. Calculate:
  1. the angle subtended at the centreof the circle by the major arc correct to the nearest 0.1⁰
  2. the area of the cross section of the tunnel correct to 2d.p.
2. Calculate: (i) the area of the shaded segments in the following diagrams. (ii) The perimeter
  (Take 3.14)
1. Radius Angle at centre Length of arc