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# Solid geometry

Pyramid
Prism
Cylinder
Cone
Sphere
Distance in the
space
Mixed problems

Examples
Definition
Main formulas
Pyramid and
its elements
Truncated
pyramid
Problems

## 3. The definition of Pyramid

A
convex polyhedron
with one face a
convex polygon (the
base) and the
vertices of the base
joined by edges to
one other vertex is
called a PYRAMID.
The
remaining faces
are all triangular.
Pyramid

## 4. The types of Pyramids

A
right-regular pyramid is one in which the
base is a regular polygon and the remaining
faces are isosceles triangles

## 5. The types of Pyramids

A
regular tetrahedron has equilateral triangles as
its faces, and so all its edges have the same
length

Vertex
Edge
Base
Lateral
face

## 9.

The elements of pyramid
F
l
h
E
β R
A
m
D
αr
γ
C
B

## 10.

The examples of Pyramid in real life
The molecule of Metan CH4
Egyptian pyramids
Louvre museum
Louvre museum
The palace of peace and reconciliation

The comparison
of world’s
pyramids

## 13. The Surface Area and Volume of an arbitrary Pyramid

1. What does the Base Area of
a pyramid depend on?
SU13
2. How to calculate the Lateral
Surface Area in a pyramid?
Lateral
surface Area
3. How to calculate the Total
Surface area of a pyramid?
Base Area

## 14. The Surface Area and Volume of an arbitrary Pyramid

Volume
1
V Abase h
3
of a pyramid:
Lateral
Surface Area of a right-regular pyramid:
(m-apothem):
1
Alat
2
P m
Tetrahedron
(a solid figure bounded by four
triangular faces. A regular tetrahedron has
equilateral triangles as its faces) a - a side:
A a
2
3
a 6
a 6
3; V
;R
; r
;
12
4
12
a
2

## 15. The definition of Truncated pyramid

A
convex polyhedron
with one face a
convex polygon (the
base) and the
vertices of the base
joined by edges to
one other vertex is
called a PYRAMID.
The
remaining faces
are all triangular.

## 16. Truncated pyramid

1. Show the lateral face area in a truncated pyramid.
2. For regular truncated pyramid
(P1 and P2 are the perimeters of
pyramid’s bases, m is an apothem):
1
Alat P1 P2 m
2
3. How to find the TSA of a
truncated pyramid?
Base2
h
TSA Alat A1 A2
m
Base1
4. Volume:
1
V h S1 S 2 S1S 2
3

3D examples
Problem #1
Problem #4
Problem #7
Problem #2
Problem #5
Problem #8
Problem #3
Problem #6
Problem #9

## 18.

Problem #1
D
C
All lateral edges in a triangular
pyramid equal 40 , the sides of the
base are 10, 10 and 12. Find the
height of the pyramid.
B
O
A
abc
R
4A
Abase p( p a)( p b)( p c)
15
4

## 19. Problem #2

D
B
A
right triangle which legs are 6
and 8 is the base of a pyramid. All
dihedral angles on the base equal
60˚. Find h.
C
O
E
A
A
r
p
DO EO tg 60 2 3
Answer : h 2 3

## 20. Problem #3

K
A
rhombus which side is 14 and
acute angle equals 60˚ is the base
of a pyramid. Dihedral angles on
the base are 45˚. Find V.
C
B
E
A
O
L
D
Abase 98 3
1
7 3
OL DE
2
2
DE 7 3
KO OL
7 3
2
1
Answer : V Abase KO 343cm3
3

## 21.

Examples
Definition
Prism and its elements
Main formulas
Problems

## 22. Examples

Astana
calcite crystal lattice
The dispersion of light and simple colors.Newton

## 23. The definition of Prism

A
convex polyhedron
with two “end” faces
that are congruent
convex polygons lying
in parallel planes in
such a way that, with
edges joining
corresponding
vertices.
The
remaining faces
are parallelograms

## 24. The types of Prism

A
right-regular prism is one in which the two end
faces are regular polygons and the remaining
faces are rectangular.

## 25. The types of Prism. Parallelepiped

A
parallelepiped is a prism in which the two
end faces (bases) are parallelograms.
A
right parallelepiped is one in which 4 lateral
faces are rectangles, otherwise its inclined.

## 26. The types of Prism

A
cube is a right
parallelepiped in
which all edges are
equal.
A
rectangular
parallelepiped is a
right parallelepiped
in which the bases
are rectangles

## 27. The types of Prism

Make
the
definitions of the
following solids

## 29. Surface area and Volume of a Prism

Lateral
Surface
Inclined Prism
Right Prism
Alat Psec l
Alat Pbase H
where Psec is a perimeter of
where Pbase is a perimeter of the
a perpendicular section and l
base and H is a height
is a length of a lateral edge
Total
Surface
Area
Volume
TSA
V Asec l
TSA
V Sbase H

## 30.

Examples
Definition
Cylinder and its elements
Main formulas
Problems