-about circle -invertor of circle -history -area of circle -example -sector -Theorem of circle -Application of circle -summary -

1. CIRCLE
DONE BY:
ANGEL ANTO ( FIRST SEMESTER)
DEPARTMENT OF COMPUTER SCIENCE
DISCRETE MATHEMATICS…

2. CIRCLE
A circle is a plain figure enclosed by a
curved line, every point on which is
equidistant from a point within, called the
centre.
2

3. William Jones
(mathematician)
• William Jones, FRS (1675 – 3 July 1749) was a
Welsh mathematician, most noted for his use of the
symbol π (the Greek letter Pi) to represent the ratio
of the circumference of a circle to its diameter.
INVENTED

4. “
The greeks considered the Egyptians as the inventors of
geometry. The scribe Ahmes, the author of the Rhind
papyrus, gives a rule for determining the area of
a circle which corresponds to π = 256 /81 or
approximately 3. 16. The first theorems relating
to circles are attributed to Thales around 650 BC.
HISTORY OF CIRCLE
4

5. DEFINITION
Circumference -
The circumference
of a circle is the
perimeter
⋄ .
Diameter - The
diameter of a circle
is longest distance
across a circle
Radius - The
radius of a circle is
the distance from
the center of the
circle to the outside
edge.
5

6. AREA OF CIRCLE
⋄ Areaofcircle= πr2
⋄ Where π = 3.142
EXAMPLE
⋄ 𝑟 =
𝑑
2
⋄ =
8
2
⋄ = 4𝑐𝑚
⋄ 𝐴 = 𝜋𝑟2
⋄ = 3.14 × 4 × 4
⋄ = 3.14 × 16
⋄ = 50.27𝑐𝑚2
6

7. ARC
A portion of the
circumference of a
circle.
𝑙𝑎 =
𝑛
360
2𝜋𝑟
A circle is 360
𝑙𝑎 =
𝑛
360
2𝜋𝑟
=
45
360
2 × 3.14 × 12
⋄ =
1
8
× 75.41
⋄ =9.43cm
ARC LENGTH
(DEGREE)
EXAMPLE
7

8. RADIAN
The angle made by taking the radius and wrapping it along the
edge of the circle.
8

9. RADIAN
FROM DEGREE TO RADIAN
Radians=
𝜋
1 8 0
× 𝑑 𝑒 𝑔 𝑟 𝑒 𝑒
FROM RADIAN TO DEGREE
Degree=
1 8 0
𝜋
× 𝑅 𝑎 𝑑 𝑖 𝑎 𝑛 𝑠
9

10. ARC LENGTH (RADIAN)
⋄ 𝑙𝑎 = 𝑟𝜃
⋄ Where r radius
⋄ 𝜃 radians
⋄ 𝑙𝑎 = 𝑟𝜃
⋄ =4.16cmx 2.5 rad
⋄ = 10.4cm
EXAMPLE
10

11. SECTOR
A sector is the part of a circle
enclosed by two radii of a
circle and their intercepted arc.
11

12. ⋄ AREA OF
SECTOR(DEGREE)
⋄ 𝐴 =
𝑛
3600 𝜋𝑟2
EXAMPLE
.𝐴 =
𝑛
3600 𝜋𝑟2
.𝐴 =
45
3600 × 3.14 × 36
.𝐴 = 14.14 𝑐𝑚2
12

13. AREA OF SECTOR
(RADIAN)
.𝐴 =
1
2
𝑟2𝜃
EXAMPLE
.𝐴 =
1
2
𝑟2𝜃
.𝐴 =
1
2
52 × 1.4
.𝐴 = 17.5𝑐𝑚2
1.4 rad
13

14. SEGMENT
The segment of a circle is the
region bounded by a chord
and the arc subtended by the
chord.
Chord of a circle is a line
segment whose ends lie on
the circle.
CHORD
14

15. SUMMARY
15

16. Theorem 1:
Prove that the tangent at any point of a circle is perpendicular to the radius
through the point of contact
Given: XY is a tangent at point P to the circle
with centre O.
To prove: OP ⊥ XY
Construction: Take a point Q on XY other than
P and join OQ
O
P Q Y
X
16

17. Proof: If point Q lies inside the circle, then XY will become a secant
and not a tangent to the circle
OQ > OP
This happens with every point on the line XY except the point P. OP
is the shortest of all the distances of the point O to the points of XY
OP ⊥ XY …[Shortest side is the perpendicular]
17

18. Theorem 2:
A line drawn through the end point of a radius and perpendicular to it, is the
tangent to the circle.
Given: A circle C(O, r) and a line APB is perpendicular to OP,
where OP is the radius.
To prove: AB is tangent at P.
Construction: Take a point Q on the line AB, different from P and
join OQ.
18

19. Proof: Since OP ⊥ AB
OP < OQ ⇒ OQ > OP
The point Q lies outside the circle.
Therefore, every point on AB, other than P, lies outside the circle.
This shows that AB meets the circle at point P.
Hence, AP is a tangent to the circle at P.
o
P B
A Q
19

20. Some examples of circles in real life are camera
lenses, pizzas, tires, Ferris wheels, rings,
steering wheels, cakes, pies, buttons and a
satellite's orbit around the Earth. Circles are
simply closed curves equidistant from a fixed
center. Circles are special ellipses that have a
single constant radius around a center.
APPLICATIONS:-
20

21. 21
Reference
 https://byjus.com/maths/circles/
 https://sites.google.com/site/circleswagger/circles-in-real-life
 https://www.google.com/search?q=William+Jones+(mathematician)&safe=active&s
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22. THANK YOU!!