HISTORY OF SOLID GEOMETRY AND

ITS CONTRIBUTION TO THE PRESENT

TIME
PROPONENTS OF SOLID GEOMETRY
 He was Greek mathematician
provided principles to thrived for
gaining knowledge in mathematical
education and made various
contributions. He is best known for
antiquity and the invention of
compound pulleys and screw pump.
He contributed so much to Solid
Geometry.
Eudoxus of Cnidus  He was a Greek astronomer who
made important contributions to
the field of geometry. He is
thought to have contributed to the
theory of proportion forms the
basis for the general account of
proportions found in book V of
Euclid's Element.
  He was one of the giants of 18th century
LEONHARD EULER
mathematics. He also made contributions
to the understanding of planar graphs. He
introduced a formula governing the
relationship between the number of edges
, vertices, and faces of a convex
polyhedron. Given such a polyhedron, the
alternating sum of vertices, edges and
faces equals a constant:

V-E+F=2
Bartel Leendert  He was a Dutch mathematician and
van der Waerden
historian of mathematics. According to
him, there are so many similarities
between the studies of Egyptians,
Babylonians, Chinese and also Indians
that he believes that these different
civilizations’ work originate from a
common source.
Cromwell
In his Polyhedra, mentions the probability
that Greek mathematicians, who liked
traveling a lot, needed proofs to decide
whether Babylonians’ methods or
Egyptian’s methods are the true ones.
  Plato, when the concern is polyhedra, is
Plato (427- 347 BC) most well known as the comments on the
five regular polyhedra, which are named
after him. Although the regular polyhedra
are called the Platonic solids, Plato was
not the first to recognize them.
Pythagoreans already knew three of them
for their regularity: the cube, the
tetrahedron (they would call it a pyramid)
and the dodecahedron.
  Theaetetus, a friend of Plato, is known to
Theaetetus
first discover the regularity of the
icosahedron and the octahedron. It
must be emphasized that these solids
were already known to people, but
Pythagoreans and Theaetetus were the
ones discovering their regularity.
Ancient Greeks believed that the physical world was
made up of four basic elements and their combinations:
fire, air, water and earth. But there still remains one
polyhedron out when four of them are assigned to the four
basic elements. Plato associated the remaining
polyhedron, the dodecahedron, to the universe, and
named a fifth element: ether.
As God brought into being the celestial virtue, the fifth
essence, and through it created the four solids . . . earth, air,
water, and fire ... so our sacred proportion gave shape to
heaven itself, in assigning to it the dodecahedron . . . the solid
of twelve pentagons, which cannot be constructed without our
sacred proportion. As the aged Plato described in
his Timaeus.

[Pacioli, L., De Divina Proportione, 1509]

 Solid geometry
is the geometry of three-dimensional Euclidean
space. It includes the measurements of volumes of
various solid figures (three-dimensional figures).

These include:
1.
2. CYLINDER
3. CONE
4. SPHERE
5. PRISM
 The Pythagoreans dealt with the regular solids, but
the pyramid, prism, cone and cylinder were not studied
until the Platonists. Eudoxus established their
measurement, proving the pyramid and cone to have
one-third the volume of a prism and cylinder on the
same base and of the same height. He was probably also
the discoverer of a proof that the volume enclosed by a
sphere is proportional to the cube of its radius.
People use solid geometry
for many purposes
including engineering and
the calculation
of perspective.
 WHAT IS
SOLID GEOMETRY?
Solid Geometry is the
Three
geometry of
Dimensions
three-dimensional space
It is called three-
dimensional, or 3D
because there are three
dimensions: width, depth
and height.
Polyhedron
A geometric object with
flat faces and straight
edges.
a polyhedron is a three -
dimensional figure made
up of sides called faces,
each face being a polygon
 Properties
Solids have properties (special things about them), such
as:
volume (think of how much water it could hold)
surface area (think of the area you would have to
paint)
how many vertices (corner points), faces and
edges they have.
 Two main types of solids:
1. Polyhedra
 (they must have flat faces)

2. Non-Polyhedra
 (when any surface is not flat)
 Polyhedra
Examples:

 Cubes and Cuboids (Volume of a Cuboid)

 Platonic Solids
 Prisms

 Pyramids
 Non-Polyhedra
Examples:
 Sphere > Torus

 Cylinder > Cone

 Base
The lowest part. The
surface that a solid
object stands on, or the
bottom line of a shape
such as a triangle or
rectangle.
 Vertices, Edges and Faces

A vertex is a corner.
An edge is a line segment between faces.
A face is a single flat surface.
 Vertices
This tetrahedron has 4 vertices.

A vertex (plural: vertices) is a point where two or more line

segments meet.
Example:
This pentagon has 5 vertices:
 Edges
This Pentagon has 5 Edges

For a polygon an edge is a line segment on the boundary joining one

vertex (corner point) to another.
This Tetrahedron has 6 Edges

For a polyhedron an edge is a line segment where two faces

meet.
 Faces
A face is any of the individual flat surfaces of a solid object.

This tetrahedron has 4 faces (there is one face you can't see)
 Sides
"Side" is not a very accurate word, because it can mean:

An edge of a polygon, or
A face of a polyhedron
 Euler's Formula
For many solid shapes the
Number of Faces
plus the Number of Vertices
minus the Number of Edges always equals 2
This can be written:
F+V−E=2
Regular Polyhedron – a solid bounded by planes whose faces are
congruent regular polygons. There are only five regular polyhedrons
namely:
a) Tetrahedron d) Dodecahedron
b) Hexahedron (cube) e) Icosahedron
c) Octahedron

a b c d e
 Regular Polyhedrons
Name Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron

Type of Face Triangle Square Triangle Pentagon Triangle

No. of faces 4 6 8 12 20

No. of Edges 6 12 12 30 30
No. of
Vertices
4 8 6 20 12
Formulas for
Vx 3
V  7.66 x 3
V  2.18 x 3
2 2
V  x3 V  x3
Volume 12 3

where: x = length of one face

Lateral Area = (number of faces) (area of one face)
 Polygons are named according to the number of sides.

Polygons
3 sides = Triangle 10 sides = Decagon
4 sides = Quadrilateral 11 sides = Undecagon
= Tetragon 12 sides = Dodecagon
5 sides = Pentagon 15 sides = Quidecagon
6 sides = Hexagon = Pentedecagon
7 sides = Heptagon 16 sides = Hexadecagon
8 sides = Octagon 20 sides = Icosagon
9 sides = Nonagon 1000 sides = Chillagon
= Enneagon
 Solids with Plane Surfaces
Polyhedron – a solid bounded by planes. The bounding planes are
referred to as the faces and the intersections of the faces are called the
edges. The intersections of the edges are called vertices.
Rectangular Parallelepiped - polyhedron with six faces which are all
rectangles.

V  abc c
A(surface)  2ab  bc  ac
b
A(lateral)  2bc  ac  a
Cube – a polyhedron with six faces which are all squares.
a
A(surface)  6a 2 A(lateral )  4a 2 V  a3

a
a
Prism – a polyhedron having two faces which are equal polygons in parallel
planes and whose other faces are parallelograms.
V  Bh
A(lateral )  PL
h=L
L h
A(surafce)  A(lateral)  2B
where: B
P = perimeter of base B
L = slant height Oblique Prism Right Prism
B = base area
 Truncated Prism
h3

h1
  heights  h5
V  B  h4
h2

 number of heights 
B

Pyramid – a polyhedron whose base is a polygon of any number of sides and whose
other faces are triangles with a common vertex.
1
V  Bh
3
A(lateral)   Afaces h

A(surface)  Alateral  B
B
 Solids with Curve Surfaces
Cylinder – a solid bounded by a closed cylindrical surface and two parallel
bases.

V  Bh  KL
A(lateral)  PK L  2rh L
K h h
A(surface)  A(lateral)  2B
B

where:
B

Pk = perimeter of right section

K = area of right section
B = base area
L = slant height
Cone – a solid bounded by a conical surface and a plane cutting all the
elements.

Right Cone L Oblique Cone

1
V  Bh h h
3
r
r
B B

Frustum of a Cone – is that portion of a cone bounded by the base and a

plane parallel to the base.
h

V  A1  A 2  A1A 2
3

A2 r
A(lateral)  R  r L L
h
where: L = slant height A1 R
 Spheres and its Families
Sphere – a solid all points on its surface are equidistant from a point within
it.
4 3
V  r r
3

A(surface)  4r 2

Spherical Lune – is that portion of a spherical surface bounded by the

halves of two great circles.

r 2
A (surface)  
90 r
where:  must be in degrees 
Spherical Zone – is that portion of a spherical surface between two parallel
planes. A spherical zone of one base has one bounding plane tangent to the
sphere.
h
A( zone)  2rh
r r r h

Spherical Segment – is that portion of a sphere bounded by a zone and the

planes of the zone’s bases.
h a
a
h 2
r r V 3r  h  r h
3 V
h 2
6

3a  3b 2  h 2  b
V
6

h 2
3a  h 2 
Spherical Wedge – is that portion of a sphere bounded by a lune and the
planes of the half circles of the lune.

r 3
V
270  r

where:  must be in degrees 

Spherical Cone – is a solid formed by the revolution of a circular sector

about its one side (radius of the circle). h
1
V  A ( zone ) r r r
3
A( zone)  2rh
A(surface)  A( zone)  A(lateralofcone)
Thank you for
listening!!!