The document provides a history and overview of circles. It discusses: 1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles. 2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi. 3) Circles can be defined analytically using equations in the standard "center-radius" form of (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the

1. Circle
Prepared by:
Joshua Cabarle
Maureen Villamora
Geraldine Reyes
Bea Nelene Que

2. PARTS AND
HISTORY OF A
CIRCLE
Prepared by : Geraldine D.
Reyes BSEd I-C

3. History

4. EUCLID (Circa 300 BC)
• Euclid was born in Alexandria, Greece. He
founded the first school of Mathematics in
Alexandria. He has been called the “Father of
High School Geometry”. Euclid wrote the book
titled “The Elements” which consists of plane and
solid Geometry. He defined a point as ” that
which has no parts,” ”a line has length without
breadth”. In modern Geometry, these terms are
left as undefined because it is impossible to state
precise definitions for all these terms.

5. Also, some of Euclid’s Theorems are now taken
as postulates because the proofs that he had
offered had Logical shortcomings.
Nevertheless, most of the theorems and much
of the development contained in a typical
High School Geometry subject today are taken
from Euclid’s Elements.

6. • The compass in this 13th century manuscript is a symbol of God's
act of Creation. Notice also the circular shape of the halo
• The etymology of the word circle is from the Greek, kirkos "a
circle," from the base ker- which means to turn or bend. The
origin of the word "circus" is closely related as well.
• The circle has been known since before the beginning of recorded
history. Natural circles would have been observed, such as the
Moon, Sun, and a short plant stalk blowing in the wind on
sand, which forms a circle shape in the sand. The circle is the basis
for the wheel, which, with related inventions such as gears, makes
much of modern civilization possible. In mathematics, the study of
the circle has helped inspire the development of
geometry, astronomy, and calculus.
• Early science, particularly geometry and astrology and
astronomy, was connected to the divine for most medieval
scholars, and many believed that there was something intrinsically
"divine" or "perfect" that could be found in circles.[citation needed]

7. • Some highlights in the history of the circle are:
• 1700 BC – The Rhind papyrus gives a method to find
the area of a circular field. The result corresponds to
256/81 (3.16049...) as an approximate value of π.
• 300 BC – Book 3 of Euclid's Elements deals with the
properties of circles.
• In Plato's Seventh Letter there is a detailed definition
and explanation of the circle. Plato explains the
perfect circle, and how it is different from any
drawing, words, definition or explanation.
• 1880 – Lindemann proves that π is
transcendental, effectively settling the millennia-old
problem of squaring the circle.

8. Analytic results
Length of circumference
• Further information: Pi
• The ratio of a circle's circumference to its
diameter is π (pi), an irrational constant that
takes the same value (approximately
3.141592654) for all circles. Thus the length
of the circumference (c) is related to the
radius (r) by
• or equivalently to the diameter (d) by

9. Circle

12. • Chord, secant, tangent, and diameter.
• Arc, sector, and segment
• The diameter of a circle is the length of a line segment whose endpoints
lie on the circle and which passes through the centre of the circle. This is
the largest distance between any two points on the circle. The diameter
of a circle is twice its radius.
• As well as referring to lengths, the terms "radius" and "diameter" can
also refer to actual line segments (respectively, a line segment from the
centre of a circle to its perimeter, and a line segment between two
points on the perimeter passing through the centre). In this sense, the
midpoint of a diameter is the centre and so it is composed of two radii.
• A chord of a circle is a line segment whose two endpoints lie on the
circle. The diameter, passing through the circle's centre, is the longest
chord in a circle. A tangent to a circle is a straight line that touches the
circle at a single point, thus guaranteeing that all tangents are
perpendicular to the radius and diameter that stem from the
corresponding contact point on the circumference. A secant is an
extended chord: a straight line cutting the circle at two points.
• An arc of a circle is any connected part of the circle's circumference. A
sector is a region bounded by two radii and an arc lying between the
radii, and a segment is a region bounded by a chord and an arc lying
between the chord's endpoints.

13. • A circle is a simple shape of Euclidean geometry consisting
of those points in a plane which are equidistant from a
given point called the centre (British English) or center
(American English). The common distance of the points of a
circle from its centre is called its radius.
• Circles are simple closed curves which divide the plane into
two regions, an interior and an exterior. In everyday
use, the term "circle" may be used interchangeably to refer
to either the boundary of the figure (also known as the
perimeter) or to the whole figure including its interior.
However, in strict technical usage, "circle" refers to the
perimeter while the interior of the circle is called a disk.
The perimeter of a circle is also known as the
circumference, especially when referring to its length.
• A circle is a special ellipse in which the two foci are
coincident. Circles are conic sections attained when a right
circular cone is intersected with a plane perpendicular to
the axis of the cone.

14. Area enclosed

15. • Area of the circle = π × area of the shaded square
• Main article: Area of a disk
• As proved by Archimedes, the area enclosed by a
circle is π multiplied by the radius squared:
• Equivalently, denoting diameter by d,
• that is, approximately 79% of the circumscribing
square (whose side is of length d).
• The circle is the plane curve enclosing the
maximum area for a given arc length. This relates
the circle to a problem in the calculus of
variations, namely the isoperimetric inequality

16. Calculating the area of a circle
A. By approximation
• We can demonstrate that the polygons shown
below in figure 1 have areas given by the
following formulas:
the square has an area equal to 2r2;
the hexagon has an area equal to approximately
2.6r2;
the octagon has an area equal to approximately
2.8r2;
the dodecagon has an area equal to 3r2.

17. B. Using the exact formula
• Area A of a circle with radius r is equal to r2.
Remember that the value of is approximately
3.14.
• So, A = × r2 = × r × r.
• To apply this formula, A and r must be
expressed in corresponding units; for
example if r is in cm, then A is in cm2.

18. II. Examples
• A. First example
• Calculating the area A of a circle with a radius of
10 cm.
• × 10 cm × 10 cm = 100 cm² 314 cm², therefore
A = 314 cm².
• B. Second example
• Calculating the area A of a circle with a diameter
of 25 m.
• The radius equals 12.5 m (half of 25 m).
• × 12.5 m × 12.5 m = 156.25 m2 491
m2, therefore A = 491 m2.

19. CIRCLE
Bea Nelene A. Que

20. Center-radius form
• A circle is the set S of all points in R2 that are
at a fixed distance r from a fixed point C.
• A circle is a locus(set) of points in a plain
equidistant from a fixed point.
• The fixed point C is called the center of the circle
and the fixed distance r is called the radius. Note
that r is the undirected distance from the center
C to any point P on the circle. Hence, the value of
r is a nonnegative real number or r>0. it thus
becomes clear that a circle with center C and
radius r is the graph of the relation.

21. S= |P(x,y)|r=|CP| (1)
• Let C (h,k) be the center of a circle and let
P(x,y) be any point on the circle. Then from
(1) and by the distance formula, we have:
(x-h) 2 + (y-k) 2 = r2
Consequently, we have proved the following
theorem.
A circle with center C(h,k) and the radius r is
the graph of the relation.

22. (x-h) 2 + (y-k) 2 = r2  center radius
form of a circle or standard form
• The center radius form of the equation of a
circle simply because it extents the coordinates (
h,k ) of the center and the radius r of the circle.
This form is also called the standard form for the
equation of a circle.
NOTE:
If the center is at the origin, then h=k=0. Hence
the standard form reduces to the form: x 2 + y 2
=r 2 .

23. Equation of Circles
Circle whose center is at the origin Circle whose center is at (h,k)
Equation: (This will be referred to as the "center- radius form".
It may also be referred to as "standard form".)
Example: Circle with center (0,0), radius 4 Equation:
Graph: Example: Circle with center (2,-5), radius 3
Graph:

24. EXAMPLE
Find the center and radius of the circle with equation
x2 - 4x + y2 - 6y + 9 = 0
SOLUTION
• In order to find the center and the radius of the circle, we first rewrite the
given equation into the standard form as given above in the definition. Put all
terms with x and x2 together and all terms with y and y2 together using
brackets.
(x2 - 4x) +( y2 - 6y) + 9 = 0
• We now complete the square within each bracket..
(x2 - 4x + 4) - 4 + ( y2 - 6y + 9) - 9 + 9 =0
(x - 2)2 + ( y - 3)2 - 4 - 9 + 9 = 0
• Simplify and write in standard form
(x - 2)2 + ( y - 3)2 = 4
(x - 2)2 + ( y - 3)2 = 22
• We now compare this equation and the standard equation to obtain.
center at C(h , k) = C(2 , 3)
and radius r = 2

25. THE GENERAL FORM
Consider again the standard form for the
equation of a circle, i.e.
(x-h) 2 + (y-k) 2 = r2
Expanding the binomials and rearranging
terms, we obtain
x2 + y2 +(-2h)x +(-2h)y+(h2 + k2 -r2 ) =0
If we let D=-2h, E=-2k, and F=h2 +k2 =r2 then
the preceding equation takes form.

26. NOTE:
 General form is an equation of the second degree in x and y (i.e. quadratic in x and y.)
 When the equation of a circle appears in "general form", it is often beneficial to convert the
equation to "center-radius" form to easily read the center coordinates and the radius for
graphing.
1. Convert into center-radius form.
We will be creating two perfect square trinomials within the equation.
• Start by grouping the x related terms together and the
y related terms together. Move any numerical
constants (plain numbers) to the other side.
• Get ready to insert the needed values for creating the
perfect square trinomials. Remember to balance both
sides of the equation.
• Find each missing value by taking half of the "middle
term" and squaring. This value will always be positive
as a result of the squaring process.
• Rewrite in factored form.
You can now read that the center of the circle is at (2, 3)
and the radius is

27. To determine the graph of convert this
equation to form (x-h) 2 + (y-k) 2 = r2 .
• If r2 > 0,then the graph is a circle with center.
• If r2 = 0,then the graph is the single point.
• If r2 < 0,then the graph is the null set.
EXAMPLE:
Determine whether the graph of the equation
x 2 + y 2 - 10 x - 8y + 32 = 0 is a circle, a
point, or the null set.

28. SOLUTION:
x 2 – y 2 - 10x – 8y + 32 = 0
(x 2 - 10x ) + (y 2 – 8y ) = -32
(x 2 - 10x + 25) + (y 2 – 8y + 16) = -32 + 25 + 16
(x - 5) 2 + (y - 4 ) 2 = 9
Since r2 = 9 > 0, then by the graph
given, equation is a circle with center ( 5, 4 )
and the radius 3.

29. Tangent to a
circle
Maureen D. Villamora
BSED IC

30. Tangent to a circle
Definition:
The word “tangent” comes from Latin
tangere, meaning to touch.
• Touching a curve at a single point or along
one line.
• In geometry, tangent circles ( also known as
kissing circles) are circles that intersect in a
single point. There are two types of tangency:
internal or external.

31. • In geometry, the tangent line (or simply the
tangent)to a curve at a given point is the
straight line that “just touches” the curve at
that point. As it passes through the point
where the tangent line and the curve
meet, or the point of tangency, the tangent
line is “going in the same direction” as the
curve, and id the sense it is the best straight-
line approximation to the curve at that point

32. • A line, a segment, or a ray that intersects a circle at
exactly one point.
• The line perpendicular drawn to a radius through the
end point of the radius is a tangent to the circle.
• A line drawn perpendicular to a tangent through the
point of contact with a circle passes through the
centre of the circle.
POINT OF TANGENCY:
The point of intersection of a circle and a tangent
to the circle.

34. CIRCLES IN THE COORDINATE
PLANE
Reported by: Joshua H. Cabarle
BSED I-C

35. THE EQUATION OF A CIRCLE
• The distance formula is used in developing an
equation for the graph of a circle
• REMEMBER:
• Every circle whose center is at the origin has
an equation of the form X 2 + Y 2=r2

36. • If the circle with equation X 2 + Y 2=r2 is
translated in the coordinate plane by moving
every point of the circle, h units horizontally
and k units vertically, the circle with radius r
and center ( h, k ) has the equation
(x - h)2 + (y - k)2 = r2
REMEMBER:
• The equation of a circle whose center is at (
h, k ) and with radius r is (x - h)2 + (y - k)2 = r2

37. IDENTIFYING THE CENTER AND
RADIUS OF A CIRCLE
• There are times when the equation of the circle
is given in the form
• x 2 + y 2=r2 + ax + by + c = 0
• This is the general equation of the circle. It can
also transformed into the standard form
(x - h)2 + (y - k)2 = r2
by completing squares. This form also gives us
information about the graph of the circle, the
location of its center, and the length of its
radius.

38. WRITING THE EQUATION OF A
CIRCLE
If the center and the radius of the circle can be
identified using the equation
(x - h)2 + (y - k)2 = r2
, the equation of the circle can be determined
if its center and the radius are given.

39. USING THE EQUATION OF A
CIRCLE
A line may intersect a circle at two points, at one
point or at no point at all.