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| Solid
Geometry |
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Prisms |
Cube
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Rectangular prism
or rectangular parallelepiped
(cuboid)
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Right triangular prism
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Regular right triangular prism
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Regular right hexagonal prism
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Pyramids |
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Regular
square pyramid
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Regular triangular pyramid
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Regular hexagonal pyramid
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Right
pyramidal frustum
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Regular Polyhedrons |
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Tetrahedron
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Octahedron
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Solids
of Revolution |
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Cylinder
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Cone
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Conical
frustum
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Sphere
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Spherical
cap |
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Spherical
segment |
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Spherical
sector |
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| Prism |
| A
prism is a polyhedron (having five or more faces) with two parallel and congruent polygonal bases, so
that |
| all cross-sections taken parallel to the bases are also congruent with the
bases, thus all lateral faces (sides) |
| are parallelograms. |
| Lateral
faces meet in line segments called lateral
edges. |
| A
right prism is one whose lateral
faces and lateral edges are perpendicular to its bases. The
lateral faces of |
| a right prism are all rectangles, and the
height of a right prism is equal to the length of its lateral
edge. |
| A
regular prism has regular polygons as bases. A regular
polygon is one that has all sides equal in length |
| and all angles
equal in measure. |
| Thus,
a right regular prism is one with
regular polygon bases and perpendicular rectangular lateral
sides. |
|
 |
| S
= 2B + Slat |
-
surface of a prism |
|
Slat = P ·
h |
-
lateral surface |
| V
= B · h |
- volume
of a prism |
| B
-area of
base, P
- perimeter of base, h
-height
of a prism |
|
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|
| Cube
|
| A
solid with six identical square faces that are mutually
perpendicular. |
|
 |
 |
|
 |
- diagonal |
| S
= 2B + Slat
= 2a2
+
4a2
= 6a2 |
-
surface |
| V
= B · h = a2
· a = a3 |
- volume |
|
|
|
| Rectangular prism or
rectangular parallelepiped (cuboid)
|
| A
solid of which the six faces are mutually perpendicular
rectangles is called a rectangular parallelepiped or
a
|
| rectangular prism.
|
|
 |
 |
- diagonal |
| S
= 2B + Slat
= 2(ab + ac + bc) |
-
surface |
| V
= B · h = a
· b · c |
-
volume |
|
|
|
| Right triangular prism
|
| A
right triangular prism is made of two triangular bases and three
rectangular faces with lateral edges |
| perpendicular to the bases. |
|
 |
| S
= 2B + Slat
= a
ha + (a + b
+ c)· h |
-
surface |
| V
= B · h = 1/2 ·
a
ha · h |
-
volume |
|
|
|
| Regular right triangular prism
|
| A
prism made of two equilateral triangular bases and three
identical rectangular sides is called a regular right |
| triangular
prism. |
|
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|
| Regular right hexagonal prism
|
| A prism made of two regular hexagonal bases and six identical rectangular sides is called a regular right |
| hexagonal prism. |
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|
Pyramids |
|
Regular
square pyramid
|
|
Regular triangular pyramid
|
|
Regular hexagonal pyramid
|
|
Right
pyramidal frustum |
|
|
| Pyramid |
| A
solid whose base is a polygon and whose lateral faces are
triangles with a common vertex (apex) is called |
| a pyramid. |
| A
right pyramid is a pyramid in which the line joining
the centroid of the base (the point of coincidence of the |
| medians) and the apex is perpendicular to the base. |
| A
regular pyramid is a right pyramid whose base is a regular polygon and
lateral faces are congruent |
| isosceles triangles. |
| The volume of a pyramid is one third of the product of the area of the base and the perpendicular distance |
| from the vertex to the base. |
| The surface
area of a pyramid: |
S
= B
+ Slat |
|
| The volume of a pyramid: |
V
= 1/3 ·
B ·
h, |
|
|
| where
B is
the base area, Slat
is the lateral surface area and h
is the height of the
pyramid. |
|
| Regular
square pyramid
|
| A
regular square pyramid has square base and lateral faces are
four congruent isosceles triangles making the |
| same angle with
the base. |
|
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| Regular triangular pyramid
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| A
regular triangular pyramid has an equilateral triangle base, and
three congruent isosceles triangles as |
| lateral faces making the same angle with
the base. |
|
 |
 |
|
| R
-the
radius of the circumcircle, r
-the radius of the incircle |
 |
-
surface |
 |
-
volume |
|
|
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| Regular hexagonal pyramid
|
| A
regular hexagonal pyramid has a regular hexagon base, and six congruent isosceles triangles as lateral |
| faces making the same angle with
the base. |
|
|
|
| Right
pyramidal frustum |
 |
| B
: B1 = (h + x)2
: x2 |
 |
 |
-
surface |
| P,
P1
- bottom
and top base perimeter |
|
 |
-
volume |
|
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|
Regular Polyhedrons |
|
Tetrahedron |
|
Octahedron |
|
| Tetrahedron |
 |
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| Octahedron |
 |
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| The equilateral triangles are faces of, the tetrahedron (4-faced), the
octahedron (8) and the icosahedron (20), |
| while the
dodecahedron
consists of 12 regular pentagons. |
|
|
Solids
of Revolution |
|
Cylinder
|
|
Cone
|
|
Conical
frustum
|
|
Sphere
|
|
Spherical
cap |
|
Spherical
segment |
|
Spherical
sector |
|
|
| Solid
of revolution |
| A
solid figure generated by revolving a line or curve (the
generator) around a fixed axis. |
|
| Cylinder |
| A cylinder is solid of revolution
generated by rotation of rectangular around one of its sides as the axis of |
|
revolution. |
|
 |
| S
= 2B + Slat
= 2· r2p
+ 2rp
· h
= 2rp·(r
+ h) |
-
surface |
| V
= B · h = r2p
· h |
-
volume |
|
|
|
| Cone
|
| A cone is solid of revolution generated
by rotation of a right triangle around one of its legs as the axis of |
| revolution. |
|
|
| S
= B + Slat
= r2p
+ rp
s = rp
(r + s) |
-
surface |
|
| V
= 1/3 ·
B · h = 1/3
r2p
· h |
-
volume |
|
|
| Conical
frustum |
 |
|
|
| Sphere
|
| A sphere is generated by rotation of
a semicircle around its diameter. |
 |
 |
|
| S
= 4p r2 |
-
surface |
| V
= 4/3p r3
|
-
volume |
|
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| Spherical cap |
 |
 |
|
 |
-
surface |
 |
-
volume |
 |
-
volume |
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| Spherical
segment |
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| Spherical
sector |
 |
 |
-
surface |
 |
-
volume |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
