Kelompok 8

Winda Mercyta
Ambar Sonya Sijabat
D. CIRCLES

 A circle is the set of the all points in a plane whose distance from a given
point is a fixed constant length. To distinguish

Circumference
secant

radius
diameter
M M
disc

d
or
ch
tangent
Circle

Angle subtended at
segment
circumference

Angle of subtended
of centre
arc sector
Between the disc bounded by the circle and the circle itself, this boundary is called
the circumference of the circle. The point equidistant from all the point on the
circle is called its centre. A straight-line segment (or interval) from the centre of a
circle to a point on the circumference is called a radius. Any interval between two
points on the circumference lies entirely inside the circle, so that the circle is a
convex figure, a line through two points on the circumference is called a secant
and the interval between them a chord. Chords containing the centre of the circle,
the so called diameters, are the longest chord in any circle. Lines that have only
one point in common with a circle are called tangent. The segment of the
circumference between two points on it is called an arc. Angles whose arms are
secants and whose vertex lies on the circumference are said to be subtended at the
circumference by the arc (or chord) between their arms. Angles whose vertex is the
centre and whose arms are radii are said to be subtended at the centre by the arc
(or chord) between their arms. The portion of the disc between two radii is called a
sector, and the portion of a sector between the chord connecting the two end-
points of the radii and the circumference of the circle is called a the segment of the
circle. ( To avoid confusion straight-line segments will be called intervals in this
section ) (Fig).

 The circumference. It is possible to gine bounds for the circumference of a

circle of diameter d by inscribing and circumscribing polygons; for example,
the circumference c i=3 d of a regular hexagon is a lower bound for the
circumference C of the circle,and the circumference c e =( 2 √ 3 ) d <3 , 47 d of the
circumscribed hexagon is an upper bound, that is :3d < c < 3,47d.

The factor by which d must be multiplied to obtain c is denoted by the greek

letter π :c=π . d . Circumference of the circle c=πd=3 πr . This number is one of
the most important and interesting mathematical constants. One can find
arbitrary accurate approximation of π by increasing the number of sides of the
polygons used. ARCHIMEDES used a 96-gon and found bounds that are still
frequently used today. His values are :
10 10
3 < π <3 ∨3,14084507< π <3,14285714.
71 70

π=3,14159 26535 89793 23846 26433 83279 50288 41971 …

 The following rough calculation shows what an accuracy of “only” 30 places of
decimals means. A system of stars that astronomers can just make visible by hour
long exposures on photographic plates using the most powerful telescope, emitted
the light that is trapped by the plate about 2000 million years ago. Since light
travels about 9,5. 1012 km per year, these stars are about 2.109. 9,5.1012 km= 1.9 . 1022
km away from the earth. The circumference of a circle with this enormous distance
as radius is c=2 πr=3 , 8 π .1022 km.

If in calculating this circumference in kilometres only the first 30 decimals

places of π are used, the error occurs in the eight place after the decimal point and
is of the order of about 2 units. That is, the error caused by disregarding further
places of π is about 20 micrometers or 0,02 mm. it is obvious that this kind of
accuracy is never required in practice. The usual approximations are π ≈3,14 for
two place of decimals, or π ≈ 3,1419 for four places.

Since π is a transcendental number, no square can be constructed by ruler and

compass whose area is equal to that of a given circle (the problem of squaring the
circle).

Area. The are of a circle can also be approximated by the areas of inscribed and
circumscribed polygons, with π occurring in the formula. The area of a circle
A=πr =π (d /2) is proportional to the square of the radius (Fig).
2 2

Area of the circle

Circumference and area of a circle

Example :

What is the circumference and the area of a circle whose diameter is 20

centimeters ?

Answer :

The circumference = c ¿ 2 π .10=20 π cm

The area = A ¿ π (10)2=100 π cm2

 Exercises :
a. Find the area and the circumference of a circle whose radius is 12 inches.
b. Find the perimeter of this figure. Use 3.14 for π

9,4 m
4,7 m

9,4 m

c.

In figure at the left, a circle is inscribed in a square one

of whose sides is 8 inches. Find the area of the shaded
region.
 d. The area of a sector of a circle is given by formula,
n 2
A= .π r
360
D C
Where n is the number of arc degress in
the arc and r is the measure of the radius
of the circle.

In figure at left, a square inscribed in a

circle has a side of 6 inches. Find the area
A B
of the region bounded by a side of the
square and its corresponding arc.
 CHAPTER IV
SOLID GEOMETRY
A. Cubes : The cube has eight rectangular solid angles, twelve edges of equal length, and is
bounded by six equal squares

In A, B, C, D, E, F, G, and H are rectangular solid angles.

AB=BC=CD=DA= AE=BF=CG=DH =EF=FG=GH =HE are edges.

ABCD, ABFE, ADHE, BCGF, CGHD, and EFGH are equal squares

Surface Area

If a model of the surface of a polyhedron is cut along in one plane to form a connected system of
bounding surfaces. This is called a net of the polyhedron. The net of the cube consist of a
connected system of six equal squares. There are different ways of arranging the squares, two of
which are shown in the figure below:
Building a model of cube from the net :

If the length of the edges of the cube are, then the area of the six squares are 6. a . a=6 a2,so that
the surface area of a cube is
2
S=6 a

Volume.

In a plane geometry the measure of the area of a figure, for example, a square or rectangle, is
defined by covering the figure withunit square. Similarly, the measure of a volume, for example,
a cube can be defined by filling the space with unit cubes. The volume of cube with edge-length
a (say 10 units) can be completely filled in this way

There are a=(10) layers, each having a=(10) rows of a=(10) unit cubes, and so altogether:
 a . a . a=a3 ( 10.10 .10=1000 ) unit cubes.

Therefore Volume of a cube V =a3

Diagonals.

There are twelve face – diagonals, for example AC

The length of AC= √ a2+ a2= √ 2 a2= √a 2

There are four space – diagonals, dor example AG

Surface area : the length of

√
AG= ( a √ 2 ) + a2= √ 2 a2 +a 2
2

¿ √ 3 a2=a √ 3

there are six diagonals planes, for example:

rectangle ACGE and the area is a √ 2 . a=a2 √ 2

B.Cuboid : The cuboid has eight rectangular solid angles and twelveedges, equal and parallel in
fours. It is bounded by three pairs of congruent rectangles lying in parallel planes.
In A, B, C, D, E, F, G, and H are rectangular solid angles.

AB=CD=EF=GH = AE , BC =AD =FG=EH , AE=BF=CG=DH are edges.

KLMN ≅ OPQR , KLPO ≅ NMQR ,∧KNRO ≅ LMQPare rectangles.

The net of the cuboid consist of a connected system of three pairs of congruent rectangles.

Try to build a model of a cuboid from the net above if the length of the edges of the cuboid are a,
b, c, then the area of the three rectangles are ab ,ac ,∧bc, so that the surface area S is given by

S=2 ab+2 ac+ 2bc

¿ 2(ab+ac +bc)

If one pair pf square faces (a = c ), then the surface area is

S=2 ab+2 a . a+2 b . a

2
¿ 4 ab+2 a