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Title: Chapter 1: Circles
Description: Your guide to GCSE mathematics for Circles, Circle Theorems and exam-style questions
Description: Your guide to GCSE mathematics for Circles, Circle Theorems and exam-style questions
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Chapter 1: Circles
The Area of a Circle:
The AREA of a circle is the space occupied by the circle
...
𝐴 = 𝜋𝑟 2
𝑑
2
𝑑
→ 𝐴 = 𝜋( )2
4
𝑏𝑢𝑡 𝑟 =
𝜋𝑑 2
∴ 𝐴 = 𝜋𝑟 =
4
2
Parts of a Circle:
A chord – this is a line segment whose endpoints lie on the circumference of the
circle
...
A segment- this is the area enclosed by a chord and an arc of a circle
...
The yellow line refers to the chord going across the
circle
...
The shaded area refers to the segment of the circle
A sector – This is an area bounded by two radii and an arc lying between the radii
...
A tangent: This is a straight line that touches the circle at a point and it is perpendicular to
the dashed line seen above in blue
...
Major Segment
The orange line represents the major arc
...
Minor Segment
Let the length of an arc = l
Let the angle formed by the arc = ϴ
∴
𝑙
𝜃
=
𝒍=
𝜋𝑑
=
360
𝝅𝒅𝜽
𝟑𝟔𝟎
=
2𝜋𝑟
360
=
𝜋𝑟
180
𝝅𝒓𝜽
𝟏𝟖𝟎
Area of a Sector:
𝜽𝟏
minor
sector
𝜽 𝟏 = the angle formed by the 𝐦𝐢𝐧𝐨𝐫 sector
𝜽 𝟐 = the angle formed by the 𝐦𝐚𝐣𝐨𝐫 sector
𝜽𝟐
major
sector
𝜽 = the angle formed by a sector
Let the area of the sector = 𝑨 𝒔
𝑨𝒔
𝜽
𝑨𝒔
𝜽
=
=
𝝅𝒓 𝟐
𝟑𝟔𝟎
𝝅𝒓 𝟐
𝟑𝟔𝟎
∴ 𝑨𝒔 =
=
=
𝝅𝒓 𝟐 𝜽
𝟑𝟔𝟎
𝒅 𝟐
𝟐
𝝅( )
𝟑𝟔𝟎
𝝅𝒅 𝟐
𝟒∗𝟑𝟔𝟎
=
𝝅𝒅 𝟐 𝜽
𝟒∗𝟑𝟔𝟎
Examples:
1
...
Take π = 3
...
(Answer
=28
...
ii) Find the circumference of the circle with (Take π = 3
...
Find the area of the sector of a circle with radius of 2 cm and an angle of 60®
...
142)
4
...
5
...
1
Proof:
2
1- Let OAC = y and OAB= x
2- Consider triangle OBC
-Since OA and OB = radius
-AOC = 180-zx = 180- (x+x)
- OAC = OCA = y because triangle OBC is an isosceles
triangle
...
1
Similiarly for triangle AOB
https://www
...
co
...
++proof&safe=active&espv=2&bi
-AOB = 180 – (y+y)=180-zx
- OAB = OBA = x
w=1920&bih=974&site=webhp&source=lnms&tbm=isch&sa=X&ved=0ahUKEwj3qnonvTPAhXkCsAKHZ2hBkYQ_AUIBigB&dpr=1#imgrc=Zi-wZUJ3RozTsM%3A
2
https://www
...
co
...
++proof&safe=active&espv=2&bi
w=1920&bih=974&site=webhp&source=lnms&tbm=isch&sa=X&ved=0ahUKEwj3qnonvTPAhXkCsAKHZ2hBkYQ_AUIBigB#imgrc=WSkKfPv5pxAOvM%3A
Title: Chapter 1: Circles
Description: Your guide to GCSE mathematics for Circles, Circle Theorems and exam-style questions
Description: Your guide to GCSE mathematics for Circles, Circle Theorems and exam-style questions