Solid
Mensuration
Mr. MARK JAVE C. GUALBERTO, RME
Lecturer I
Basic Definition of Solid Mensuration
SOLID GEOMETRY
Solid Geometry (also known as Solid Mensuration) is the study of various solids. It is the study of the
measure of volume, area, height, length, and many more. This subject is used extensively in the practice of
engineering. The knowledge of this subject is a necessity to engineers and other construction professionals
in any project construction.
In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean
space (3D geometry).
Three-dimensional space (also: 3-space or, rarely, tri-
dimensional space) is a geometric setting in which three
values (called parameters) are required to determine the
position of an element (i.e., point). This is the informal
meaning of the term dimension.
In physics and mathematics, a sequence of n numbers can be
understood as a location in n-dimensional space. When n = 3,
the set of all such locations is called three-
dimensional Euclidean space (or simply Euclidean space
when the context is clear). It is commonly represented by the
symbol ℝ3. This serves as a three-parameter model of the
physical universe (that is, the spatial part, without
considering time), in which all known matter exists. While
this space remains the most compelling and useful way to model the world as it is experienced, it is only
one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example,
when the three values refer to measurements in different directions (coordinates), any three directions can
be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore,
in this case, these three values can be labeled by any combination of three chosen from the
terms width, height, depth, and length.
Stereometry deals with the measurements of volumes of various solid
figures (three-dimensional figures), including pyramids, prisms and
other polyhedrons; cylinders; cones; truncated cones; and balls bounded
by spheres.
History
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.
Euclid's account of spherical geometry is in his Elements volumes XI–XIII containing solid
geometry, and in his lesser-known work the Phaenomena, which includes 25 geometric propositions. The
actual discoveries were often made by others. The Pythagoreans dealt with the regular solids, like the cube
and the sphere.
Euclidean geometry is a mathematical system
attributed to Alexandrian Greek mathematician Euclid, which
he described in his textbook on geometry: the Elements.
Euclid's method consists in assuming a small set of intuitively
appealing axioms, and deducing many
other propositions (theorems) from these. Although many of
Euclid's results had been stated by earlier
mathematicians, Euclid was the first to show how these
propositions could fit into a
comprehensive deductive and logical
system. The Elements begins with plane geometry, still
taught in secondary school (high school) as the first axiomatic
system and the first examples of formal proof. It goes on to
the solid geometry of three dimensions. Much of
the Elements states results of what are now
called algebra and number theory, explained in geometrical
language.
Euclid (c.330-275 BCE, fl. c.300 BCE)
Solid figures (2D)
Shape Area (Square units) Perimeter (units) Figure
Square a2 4a
Rectangle l×w 2 ( l + w)
Circle πr2 2πr
Isosceles Triangle ½×b×h 2a + b
Equilateral Triangle (√3/4) × a2 3a
Shape Area (Square units) Perimeter (units) Figure
Right Angle Triangle ½×b×h b + hypotenuse + h
Rhombus ½ × d1 × d2 4 × side
Parallelogram b×h 2(h+b)
Trapezoid P= a+b+c+d
Trapezium ½ h(a+b) a+b+c+d
Mensuration Formulas for 3D Shapes
Volume Curved Surface Area (CSA) Total Surface
(Cubic or Lateral Surface Area Area (TSA)
Shape units) (LSA) (Square units) (Square units) Figure
Cube a3 LSA = 4 a2 6 a2
Cuboid l×b×h LSA = 2h(l + b) 2 (lb +bh +hl)
Sphere (4/3) π r3 4 π r2 4 π r2
Hemisphere (⅔) π r3 2πr2 3πr2
Volume Curved Surface Area (CSA) Total Surface
(Cubic or Lateral Surface Area Area (TSA)
Shape units) (LSA) (Square units) (Square units) Figure
Cylinder πr2h 2π r h 2πrh + 2πr2
Cone (⅓) π r2 h πrh πr (r + h)
Other Solid Figures
Figure Definitions Images
A polyhedron with six faces (hexahedron), each of
which is a parallelogram
Parallelepiped
A hexahedron with three pairs of parallel faces
A prism of which the base is a parallelogram
A parallelepiped where all edges are the same
length
Rhombohedron
A cube, except that its faces are not squares
but rhombi
Small stellated
Flat polygonal faces, straight edges and sharp corners
Polyhedron dodecahedron
or vertices
Toroidal polyhedron
Tetrahedron
Uniform Regular polygons as faces and is vertex-transitive (i.e.,
polyhedron there is an isometry mapping any vertex onto any other)
Dodecahedron
A polyhedron comprising an n-sided polygonal base, a
second base which is a translated copy (rigidly moved
Prism without rotation) of the first,
and n other faces (necessarily all parallelograms)
joining corresponding sides of the two bases
Tapers smoothly from a flat base (frequently, though not
Cone
necessarily, circular) to a point called the apex or vertex
A right circular cone and an
oblique circular cone
A solid
elliptic
cylinder
Cylinder Straight parallel sides and a circular or oval cross section
A right and
an oblique
circular cylinder
Examples of ellipsoids with
A surface that may be obtained from a sphere by
Ellipsoid deforming it by means of directional scalings, or more equation
generally, of an affine transformation sphere (top, a=b=c=4),
spheroid (bottom left,
a=b=5, c=3),
tri-axial ellipsoid (bottom
right, a=4.5, b=6, c=3)
A lens (or less than half of a circular arc) rotated about
Lemon an axis passing through the endpoints of the lens (or
arc)[6]
A surface that is generated by rotating
Hyperboloid
a hyperbola around one of its principal axes
