1. THREE DIMENSION A . The Kinds Of Three Dimension Geometric Object
B . Position Point , Line , and Sector
C . Intersection Line Two Sectors and Penetrate Line and Sector
D . Drawing The Geometric Solids
E . Postulat In Geometry Solid
F . Affinity Axis
G . The area and volume of geometrical object
H . Projection
I . The Distance Between Points,lanes and planes In the Space Figure
J . The Angle of The space Figure
HOMEWORK

2. A. The Kinds Of Three Dimension Geometric Objects
CUBE
CUBOID
CUBE 2

3. CONE
CYLINDER
PYRAMID
PRISM
SPHERE

4. B. Position point,line,and sector/PLANE
A . Position point and line
Point on the line
Point outside the line l A g A

5. (i) parallel : the both lines haven’t intersection point.
B . Position line and line
1. On the sector
(i) parallel : the both lines haven’t intersection point. l g
l // g

6. (ii) intersection : the both lines have one intersection point.g

7. 2. Not on the sector
(i) Crossing : the both lines haven’t intersection point. g l

8. C. Position Line And Sector
1. Attached : minimum there are 2 point on the sector α l

9. 2. Parallel : line and sector haven’t intersection pointsg l
g // l α

10. 3. Intersect : line and sector have one intersection point.α

11. d. Position sector and sector
Parallel : the both sector haven’t intersection points. l g α l1 g1 β

12. 2. Intersection : the both sectors have 1 intersection line.α
(α,β) Is called intersection line between sector α and sector β
(α,β) β

13. 3. Attached β α

14. Examples
Determine the positions points, lines, and sectors at the figure cube below :
AC …… BD
CP …… QE
EP …… CQ
AP …… CG
FQ…… DH
BP …… CD
CG …… AD
AP …… BC
AG …… PQ
B …… AC
AE……BDHF
AG …… ACGE
PD ……CDHG
BD ……ADHE
GE ……ABCD
DF ……BCGF
BD …… AFH
HP ……CDHG
GQ……ABFE
EF …… EFGH
ADHE …… BCGR
ACGE …… BDHF
ACP …… ACGE
GEQ …… ABCD
AFH …… BDG
BCHE …… ACGE
ABCD …… AFH
DHF …… BDF
P …… HF
C …… AQ
D …… BF
P …… ACGE
G …… EFH
A …… BDHF

15. C. Intersection line two sectors and penetrate point line and sector
Draw intersection line two sectors :
Determine two intersection points from the both sectors.
And then connected of that two intersection points (is called intersection line two sectors)

16. Examples : 1. Make intersection line of two sectors below :
a. ACGE and BDHF
b. ACH and BDG

17. c. ABGH and AFP
d. ACP and BCHE

18. answer

19. b. Draw intersection point between the line and sector
(i) Make the sector which pass through the line that given that
(ii) Find the intersection line g of the both sectors
Find the intersection point of intersection line and the line that given that

20. example
Determine the intersection point of the line and sector at the cube ABCD.EFGH.P
is lengthen of AE so AE:EP=2:1
a. DF and ACH
b. AG and CDEF
c. GP and BDHF

21. D . Drawing The Geometric Solids
The kinds of terms in drawing the solid consist of :
Frontal Side
Is the side which parallel to the drawing side/plane or the side which are draw according to the true measurement.
Frontal Line
Is the lines which be located at the frontal side
Orthogonal Side
Is the side which perpendicular to the frontal side
Orthogonal Line
Is the line which perpendicular to the frontal side
Subside Angle
Is an angle which can be determined from horizontal frontal line to the right and a orthogonal line to the back
Orthogonal Ratio
Is an comparison between the length of orthogonal line in the figure with length of orthogonal line in an actual
The orthogonal line ratio = the length orthogonal line in figure
the length orthogonal line an actual

22. Example
1. Draw the cube ABCD.EFGH with dimension 5 cm, BCGF as the frontal side, BC as horizontal line, a subside angle is 60˚ and orthogonal radio is 3/4 .
2. Draw the cube ABCD.EFGH with dimension 4 cm, ACMK is a frontal side. AC is horizontal, a subside angle is 135˚ and
orthogonal ratio is ½.
3. As like no. 1 dimension 4.5 cm, ABGH as frontal sides, AB is horizontal line, a subside angle is 30˚ and orthogonal ratio 2/3.
4. As like no. 1 dimension CDGH, CD horizontal line.
5. a.Draw cuboid ABCD.EFGH with dimension AB = 4 cm, AD = 3 cm, AE = 5 cm, ABFE as frontal line, AB is horizontal line, a subside angle 150˚ and orthogonal ratio is 3/5.
b. As like no 5. frontal side BCGF, BC horizontal line, subside angle 450, orthogonal ratio ¾.
6. Draw the regular pyramid TABCD, AB = 4 cm, the length of it 5 cm, TBD frontal side and BD horizontal line, a subside angle is 450 and orthogonal ratio is 2/3.
7. As like no 6. P is center point of AD and Q center point of BTPQ as frontal side, PQ is horizontal line, a subside angle is 1200 and orthogonal ratio is ½.

24. E . POSTULAT IN GEOMETRY SOLID
A line can be formed through two points B A
A sector can be formed through three points which not collinear (segaris ) B A C

25. A sector can be formed through 2 lines which parallel
A sector can be formed through 2 lines which intersection l g

26. F . AFFINITY AXIS
The affinity axis can be used for draw a slices sector with geometric object
The affinty axis is intersection line between slices sector with geometric object

27. CROSS SECTION OF GEOMETRIC SOLID
Definition
Cross section of geometric solid is intersection of a plane certenly and the solid
Affinity axis is an intersecting line between a cross section of geometric soild and the solid base plane

28. Example
The cube ABCD EFGH with edge 4 cm. P is center point AE, R center point CG, draw a cross section passes through point P, H and R with that cube and evaluate the area of that cross section
Answer: A B C D E F G H P L R Y X
Aff axis

29. Connect H and P point until intersect AD line in X point B C D E F G H P L R Y X
Aff axis
Algorithm
Connect H and P point until intersect AD line in X point
Connect H and R point until intersect CD line in Y point
Connect X and Y point and intersect the cube in B point
Connect B and P point and also B and R point
We get a plane HPBR is called cross section of the cube
HPBR in the form rhombus so the area of HPBR is :
L = ½ (HB).(PR)  HP = 43
= ½ 43 42 PR = 42
= 86 cm2

30. Draw a cross section pyramid T.ABCD with PQR plane in the figure below.
Answer: B C D A T S P R Q U
Aff axis
PUQSR is cross section

31. Draw a cross section of geometric solid below! a) b)
Exercise 7
Draw a cross section of geometric solid below!
a) b) A B C D E F G H K M L A B C D E F G H Q P R

32. c) d) A B C D T R Q P A B C D E F G H K M F M K e) D E C L A B

33. Exercise 8
1. Draw a slice of sector of cube ABCD, EFGH which through the point
N  NH : ND = 1 : 3
K  EK : KA = 1 : 2
L  PL : LB = 1 : 2
If the length of slice of cube 4 cm. A B C D E F G H K L M N
a slice of sector
Afinity axis

34. 2. Draw a slice of sector which through the point H, K, LB C D E F G H K L M N X Y Z
Afinity axis

35. 3. As like no. 2 which through the point K, L, MB C D E F G H K L M N

36. 4. As like no.2 which through the point E,K,LB C D E F G H K M L
AFFINITY AXIS

37. G . The area and volume of geometrical object
CUBE
If the length of edge of the cube is r,then :
The length of side/edge diagonal =r √2
The length of space diagonal =r √3
The area of sector diagonal =r2 √2
The area of the cube =6r2
Volume of the cube =r3

38. CUBOID If the length of cuboid =l,width =w and height =h,then :
The length of space diagonal =
The area of cuboid= L=2(lw+lh+wh)
Volume of the cuboid =V=lwh 38

39. UPRIGHT PRISM The length of blanket prism = perimeter of base x h
The area of the prism =2x the area of base + the area of blanket
Volume = the area of base x height  

40. OBLIQUE PRISM
The area of blanket prism = perimeter of right slices x h
The area of prism = 2x area of right slices + the area of blanket prism
Volume = the area of right slices x h 

41. PYRAMID The volume of pyramid= 1/3 x area of base x h
The area of blanket pyramid = perimeter of base x apotheme
The area of pyramid = the area of base+area of blanket

42. CONE The area of blanket cone = π R A(R= rhombus ; A= Apotheme)
The area of cone=the area of base = area of blanket=πr2 +πrA = πr(r+A)
The volume of cone = 1/3 πr2h

43. CYLINDER The area of cylinder = 2 π r2 + 2πrh
The area of blanket = 2 π rh
Volume of the cylinder =πr2h

44. SPHERE/BALL
THE AREA OF SPHERE = 4πr2
Volume of the sphere=4/3 π r3

45. The length of edge of the cube 4cm.Determine:
example
The length of edge of the cube 4cm.Determine:
The length of side diagonal
The length of spece diagonal
The area of sector diagonal
The area of cube
Volume of cube
The length of cuboid is 8cm,width=4cm,height=5cm.Determine:
The length of base side diagonal
The length of space diagonal
The area of cuboid
Volume of the cuboid

46. The upright prism,the base of prism has the form rhombus with the length diagonal 12cm and 16 cm.If the area it’s prism 400cm2.Determine :
area of base
Altitude
The area of prism
Volume of prism
Known a cone with base radius 5cm and altitude 8cm.Determine the area and volume of cone
The radius of sphere is 14 cm.Determine the area and volume of the sphere!

47. The radius of cylinder is 6 cm and the altitude of it is 10 cm
The radius of cylinder is 6 cm and the altitude of it is 10 cm.Determine
The area of cylinder
Volume of cylinder

48. w = 4 cm h = 5 cm answer a.4√2 b. 4√3 c.42√2 cm2 d.6.42 cm2 e. 43=64cm
l =8cm
w = 4 cm
h = 5 cm

49. a. The Length of base diagonal =
b. The length of space diagonal = d
c. Volume of cuboid =l.w.h
= =160 cm3
d. The area of cuboid =2.(l.w+l.h+wh)
= 2.( )
=2.92
=184 cm2

50. a . Area of base = ½ x 12 x 16 = 96 cm 2
c. area of prism = 2(96+400) = 992 cm 2
b. perimeter = 10 x 4 = 40 cm 2
h =area blanket : perimeter
=400 : 40
10 cm
d. V= L base x h
= 96 x 5 =480 cm3

51. 4. r = 5 cm
h = 8 cm
a. L cone = π r2 + π r.a
= π √89
=25 π + 25 √89
= (25+25√89) π cm2
b. V = 1/3.L base.h
=1/3. π
= 200/3 π cm3

52. 5. r = 14 cm
a. V sphere = 4/3π
= ( )/3 π cm3
b. L sphere = 4 π 14.14
= 784 cm2

53. 6 . r= 6 cm
h= 10 cm
a). L = 2 x L base + 2πh
=2.π π.6.10
= 2.36π + 120π
= 72π + 120π
=192 cm2
b). V = La x h
=π.62 x 10
= 360π cm3

54. Homework The length of space diagonal of the cube =√125.Determine
The length of side diagonal
The area of sector
The area of cube
Volume the cube
The area of the cube 108 cm2.Determine
The length of space diagonal
The area of sector diagonal
Volume
The comparison of the length,width and height cuboid are 4:3:1.The area 152 cm2.Determine volume of it!

55. The triangle upright prism ABCD
The triangle upright prism ABCD.DEF is regular with the length of edge of base sector = 2cm and the length of vertical edge =3 cm.Determine
The area of base sector
The area of blanket
The area of prism
Volume of prism
Given that hexagonal prism is regular the length of base edge is 6 cm,the length of vertical edge is 8 cm.Deter mine the area and volume of prism!
The regular triangle prism with volume 300 cm3 and the length of vertical edge = 4√3 cm.Determine:
The length oif base edge

56. In the cube ABCD. EFGH,find the ratio in volume of pyramid T
In the cube ABCD.EFGH,find the ratio in volume of pyramid T.ABCD to volume of the cube!
In the cube EFGH, O is centre point of the cube.Determine the area and volume of rectangular pyramid O.ABCD in side of the cube,if the length of edge of the cube is 4 cm
The regular triangle upright pyramid T.ABC has the length base it 6 cm and vertical it 5 cm,Determine :
The area of base pyramid
The area of sector T.AB
The area of pyramid
The altitude of pyramid
Volume of pyramid

57. If V1 is volume of the largest of cone inscribe in cylinder V
If V1 is volume of the largest of cone inscribe in cylinder V.Determine the ratio of V1:V !
The regular rectangular pyramid T.ABCD,the length of base it 8 cm,volume ofpyramid 146 1/3 cm3.Determine
The altitude of pyramid
The area of blanket pyramid
The area of pyramid

58. The figure below showes the pyramid T
The figure below showes the pyramid T.ABCD with TA=TB=TC=TD=13 cm, and of base rectangle ABCD .The length AB=BC=8 cm determine :
The area of base P
The area of blanket P
The Area of pyramid
The volume of pyramid

60. 1. Projection of point to line
Projection of A point to g line can be determined by : draw the line l from A point which perpendicular with g line and intersect in A’ point
A’ is the result projection of A A A1 g l

61. 2. Projection of point to plane
Projection of A point to  plane can be determined by : draw the g line passes through A point and perpendicular with  plane and intersect in A’  g A A1

62. 3. Projection of line to plane
Projection of g line to  plane can be determined by :
Take 2 points at g line (suppose A and B point)
Draw the line passes through A and B point and perpendicular with  plane and intersect in A’ and B’ point
Connected the both points A’ and B’ (given name g’ line)  g1 A A1 g B B1

63. I . THE DISTANCE BETWEEN POINTS, LINES AND PLANE IN THE SPACE FIGURE

64. 1. The Distance Between Two Points
The distance between 2 points A and B is the length of line from A to B point
d = |AB| A B d

65. 2. The Distance Between a point to a line
The distance between A point to g line can be determined by :
Draw the line l passes through A point and perpendicular g line and intersect in A’ point. l
d = |AA’| A g A1

66. 3. The Distance Between a point to a plane
The distance between A point to  plane can be determined by :
Draw the line g line passes through A point and perpendicular to  plane and intersect in A’. A
d = |AA’| g A1 

67. 4. The Distance Between two parallel line
The distance between g and l line which parallel can be determined by :
Draw the l line which perpendicular with g and l line and intersect g line at A and intersect l line at B
AB is called the distance of the both line that paralel. h A B g l

68. 5. The Distance Between a line to a plane that parallel
The distance between g line to  plane that parallel can be determined by :
Take A point at g line
Draw l line that perpendicular with  plane and intersect in A’ point
Draw g’ line passes through A’ point l g A
d = |AA’| g1 A’ 

69. 6. The Distance Between Two Parallel Plane
The distance between two parallel planes  and  can be determined by :
Take A point at  sector
Draw l line passes through A point and perpendicular to  plane, and intersect in A’ A 
d = |AA’| A’ 

70. 7. The Distance Between Two Crossing lines
The distance between two crossing lines g and h can be determined by :
Draw  plane passes through h line and parallel to g line
g line is projected to  plane and intersect h line at A point (give name g’)
A point is projected to g line at B point so the distance of 2 crossing line is segment AB g B
d = |AB| g1 A  h

71. EXAMPLE
In the cube ABCD EFGH with the length edge a cm, determine the distance of :
C to F
B to H
G to Q (Q is center point of AC and BD)
C to DG
E to PA (P is center point of HF and EG)
Q to FE
D to ABFE
E to BDG
A to BDG
EH to BC
AE to DF
BG to ADHE
EF to ABGH
BDG to AFH
XYZ to AFH (X is center point of EH, Y is center point of EF and Z is center point of AE) H G X P N M E F Y T R S Z L D C Q A B

72. Answer CF = a √2 cm BH = a √3 cm GQ = = C to PG = ½ CH = ½ a2 cm
E to PA = 1/3 EC = 1/3 a3 cm
QY =
AD = a cm
EE’ = 2/3 EC = 2/3 a3 H G X P N M E F Y T S R cm a 5 2 4 1 6 ) / ( FY FQ - Z L D C Q A B

73. A B C D E F G H K M N P X Y Q R T S A’ A to BDG BE = a2 cm
ZR = ½ AC = ½ a2 cm
AB = a cm
F to ABGH = ½ FC = ½ a2 cm
SR = 1/3 EC = 1/3 a3 cm
TS = 1/6 EC = 1/6 a 3 cm

74. EXERCISE 5
Known Cuboid ABCD EFGH with dimension AB=6 cm, AD=2 cm and AE = 3 cm. P point is center point AC and Q the lengthen of EH with ratio EQ : HQ = 2 : 1. find the distance to Q.
If T is center point CD in the cuboid ABCD EFGH with dimension AB=BC=4cm and AE=2cm then determine the distance T point to HB
The cube ABCD EFGH, P is center point EG, Q is center point AC and HQ=62cm. Determine the distance P to ACH
If R is center point EH and S is center point BC in the cube ABCD EFGH with the length edge 6cm then determine the distance BR line to SH line
Known the right prism ABCD EFGH with edge AB=1cm, AD=2 cm and AE=4cm, P is center point BF, determine the distance of the line AD and HP
Dimension of cuboid PQRS KLMN is PQ=2m, PS=4m and PK=6m. A is center point PS, determine the distance between RS with the plane that pass through KL and A
Determine the distance between BDG and XYZ in the cube ABCD EFGH with edge 4cm, if X center point EH, Y center point EF and Z center point AG

75. The Angle Of The space Figure
1. The angle between two lines
a. Angle between two intersect lines
 is the angle which formed by g and h line that intersect each other
Notation :
 = <(g,h)

76. (ii) The angle which formed by g’ and h line is 
b. Angle between two crossing line
(i) Draw g’ line that parallel with g line and intersect h line in A point
(ii) The angle which formed by g’ and h line is 
Notation :
 = < (g, h) = < (g’, h)

77. 2. The angle between line and plane that intersect
The angle of g line and  plane that intersect in B point can be determined by :
(i) Take A point at g line
(ii) Draw l line passes through A point which perpendicular with  plane and intersect in A’
(iii) Connect B to A’ point (give name g’)
Notation :
 = <(g,  ) = <(g, g’)

78. 3. The angle between two intersect plane
The angle between  and  plane that intersect at (, ) can be determined by :
(i) Take A point at (, )
(ii) draw the line AB at  plane and AC at  plane
(iii) AB and AC line forms the  angle
Notation :
 = < (, ) = < (AB, AC)

79. Example :
1. In the cube ABCD.EFGH with edge 4 cm, determine the angle between : a. CF and BG d. AC and BDG
b. BG and DG e. CE and AFH
c. AD and FC f. BDE and BDG
Answer :
a) <(CF,BG) = 90o
b) <(BG,DG) = 60o
c) <(AD,FC) = < (BC,FC)
= <(BCF) = 45

80. d) <(AC,BDG) = < COG
Tan < COG = CG/ OC
= a/a
= ½
< COG =
e) <(CE,AFH) = 90
f) <(BDE,BDG) = <(EO,OG)= <EOG = 

81. EO=GO ; EG = =
= cm
Cos  =
= 1/3
 =

82. 2. Pyramid T. ABCD with base plane is square
2. Pyramid T.ABCD with base plane is square. If the length base edge 4 cm and altitude of pyramid is cm, determine the value of sin . ( is the angle between TAD and ABCD)
Answer :
TO= cm
TE= =
= =
Sin= TO/TE
= 2√3/4
= 1/2 √3

83. Exercise 6
Known the cube ABCD.EFGH with edge 8cm. Determine the angle between :
a) ABFE and ABGH
b) BF and ACF
c) AF and HD
Tetrahedron A.BCD with E is center point BC, if the angle between ABC and BCD is . Determine the value of cos 
PQRS is a rhombus lies on horizontal plane. Length of its diagonal are PR = 16cm and QS = 12cm. Determine the angles between planes :
a) TQS and PQRS
b) TRQ and TRS

84. 4. From figure beside ABC is
From figure beside ABC is equilateral triangle, if  is the angle between DAB and CAB planes find tan 
Pyramid T.ABCD is regular pyramid. The length AB=BC=3cm and TA=TB=TC=TD=√6cm
a) Determine the angle between TB line and ABCD plane
b) Determine the value of sin  if  is the angle between TBC and ABCD plane