MATH 405
SOLID MENSURATION
Prepared by:
Engr. Raymond Calderon
Solids for which V= ( Area of the base)(height)
also called Rectangular
Parallelepiped
Solids - is any limited portion of space,
bounded by surfaces
Section - a plane figure cut from the solid by
passing a plane through it
Solid
Plane M
Section formed
is the plane figure
ABCDE
Polyhedron - is a solid bounded by planes
Edges - intersection of a bounding plane
Faces - portions of the bounding plane
included by the edges
Vertices - intersection of the edges
Cavalieri’s Theorem
Volume Theorem
Cube
Properties: 1. The three dimensions of a cube are equal to each
other. Therefore, all edges are equal.
2. All the faces of a cube are congruent squares.
Sample Problems
How much material was used in the manufacture of 24,000
celluloid dice, if each die has an edge of 1/4 inch?
Sample Problems
A vegetable bin built in the form of a cube with an edge of 6 ft.
is divided by a vertical partition which passes through two
diagonally opposite edges. Find the lateral surface of either
compartment.
Solution:
In this figure,
𝐴1 is a square with dimension
of 6in. By 6in.
𝐴2 is a rectangle with
dimension of 6in. by ‘d’in.
Solve first the diagonal using
pythagorean theorem.
d
6’
6’
Rectangular Parallelepiped
Properties: 1. The parallel edges of a rectangular parallelepiped are
equal.
2. The opposite lateral faces of a rectangular
parallelepiped are equal and parallel.
3. Any two opposite faces of a rectangular parallelepiped
may be taken as the base.
Sample Problem
Cylinder
Properties: 1. The bounding cylindrical surface of a cylinder is called
the lateral surface, and the two bounding parallel
planes are called the bases.
2. The bases of a cylinder are equal.
3. The altitude of a cylinder is the perpendicular
distance between the bases.
4. The sections of a cylinder made by two parallel
planes, neither of which cuts a base and both of which cut an
element, are congruent.
5. Every section of a cylinder parallel to the base has an
area equal to that of the base.
Cylinder
Sample Problem
ℎ
C = 2𝜋𝑟
Prism
Properties: 1. The bases are congruent polygons: lateral area is the
sum of the areas of the remaining faces.
2. The intersections of the lateral faces are called lateral
edges. These lateral edges are equal and parallel.
3. The sections of a prism made by parallel planes cutting
all the lateral edges are congruent polygons.
4. The altitude of a prism is a section perpendicular to its
bases; its lateral faces are rectangles.
5. A cross section of a prism is a section perpendicular to
the lateral edges.
6. A right prism is a prism whose lateral edges are
perpendicular to its bases; its lateral faces are rectangles.
Prism
Sample Problem
In this figure, the bases are in the shape of a trapezoid.