1. Solid Mensuration
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Frustum of a Pyramid
Frustum of a Right Circular Cone
Prismatoid
Truncated Prism
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2. Meryl Mae R. Nelmida
UNIT I
Frustum of a Pyramid
UNIT II
Frustum of a Right Circular Cone
UNIT III
Prismatoid
UNIT IV

3. Truncated Prism
UNIT I
FRUSTUM OF A REGULAR PYRAMID
If a pyramid is cut by a plane parallel to its base,
two solids are formed. (see fig. 1) The solid above the
cutting plane is a pyramid which is similar to the
original pyramid and the other solid formed is a frustum
of the original pyramid. In general, a frustum of a
pyramid is that portion of the pyramid between its base
and a section parallel to the base. The frustum of a
regular pyramid is also called pyramidal frustum.
Fig. 2
Fig. 1

4. Note: figure 2 represents the unfold of a frustum of a
pyramid
Properties:
• The bases of the frustum are the base of the original
pyramid and the base of the parallel section.
• The altitude/height of the frustum is the
perpendicular distance between its bases.
• The lateral faces of a frustum of a pyramid are
trapezoids.
• If the frustum is cut from a regular pyramid, then
its lateral edges are equal and its lateral faces are
congruent isosceles trapezoids.
• The slant height of the frustum of the regular
pyramid is the altitude of a lateral face.
• The bases of a frustum of a regular pyramid are
similar regular polygons. If these polygons become
equal, the frustum will become prism.
Figure 3 represents the frustum of a regular pyramid.
MNPQR and M’N’P’Q’R’ are its bases; AA’ is its altitude
and SS’ is the slant height. The segments MM’, NN’, PP’, …
are the lateral edges; MN, NP, PO, … are the lower edges;
M’N’, N’P, P’Q’, .. are the upper base edges; and MNN’M’,
NPP’M’, PQQ’P’, … are the lateral faces. Note that
relative to the frustum of a pyramid, five important line
segments are involved, namely:
1. Altitude

5. 2. Slant height
3. Lateral edge
4. Lower base edge
5. Upper base edge
Fig. 3
The lateral area S of the frustum of a regular
pyramid is equal to one-half of the product of the slant
height l and the sum of the perimeters (p1 and p2) of the
bases. In symbol,
Eqn. 1
The total area of the frustum of a regular pyramid is
the sum of the lateral area and the areas of the bases.
The volume V of the frustum of a regular pyramid
whose bases are b and B ( B > b) and with the altitude h
is given by

6. Eqn. 2
In words, the volume of the frustum of a regular
pyramid is equal to one-third the product of its altitude
and the sum of the upper bases, the lowers base, and the
mean proportional between the bases. To prove, consider
the pyramid P-MNQR in Figure 4.

7. Fig. 4
Let
H = LP + altitude of pyramid P-MNRQ
H = LL’ = altitude of the frustum with bases MNRQ and
M’N’R’Q’
b = area of the upper base M’N’R’Q’
B = area of the lower base MNRQ
V = volume of the frustum P-MNRQ
V1 = volume of the pyramid P-M’N’R’Q’
V2 = volume of the pyramid P-MNRQ
Then

8. By the equation Volume = (B × h)/3
Substituting (2) and (3) in (1) and rearranging the terms,
we get
s = l2
S L2
Also, by the equation
which states that the area (s,S) of similar
surfaces have the same ratio as the squares of any
two corresponding lines.
Or solving for H, we obtain

9. Substituting (6) in (40 and simplifying, we get Equation 2
which is
EXAMPLE:
1. Find the volume of the frustum of a regular square
pyramid whose altitude is 10 cm and whose base edges
are 4 cm and 8 cm.
Solution:
We have the following data based on the given:

10. b = 42 = 16
B = 82 = 64
H = 10
Then by Equation 2,
2. Calculate the lateral area, surface area and volume
of the truncated square pyramid whose larger base
edge is 24, smaller base edge is 14 cm and whose
lateral edge is 13 cm.
h2 = 132 - 52 = 12 cm
p1 = 24 * 4 = 96 cm
P2 = 14 * 4 = 56 cm

12. UNIT II
FRUSTUM OF A RIGHT CIRCULAR CONE
The frustum of a right circular cone is that portion
between the base and a section parallel to the base of the
cone. The terms slant height and altitude are used in the
same sense as with the frustum of a regular pyramid.
Fig. 5
Properties:

13. • The altitude of a frustum of a right circular cone is
the perpendicular distance between the two bases.
• All the elements of a frustum of a right circular
cone are equal.
•
In figure 5, we have a frustum of a right circular
cone with slant height l, altitude h, lower base radius R
and upper base radius r. it is proved in elementary solid
geometry that the lateral area of the frustum of a right
circular cone is equal to one-half the product of the sum
of the circumferences of its bases and the slant height.
That is,
Eqn. 3
Where:
c = circumference of the upper base
C = circumference of the lower base
l = slant height of the cone
S = lateral area
But c = 2πr and C = 2πr. Substituting these values,
we get
Eqn. 3.1
Where:

14. r = upper base radius
R = lower base radius
l = slant height of the cone
The volume of the frustum of a circular cone is used
in the same sense as with the volume of a regular pyramid.
That is,
But for a right circular cone, b = πr2 and B = πR2.
Substituting these values in the above equation, we get
=
Eqn. 4
Where:
V = volume of frustum
h = altitude of the frustum
r = upper base radius
R = lower base radius
EXAMPLE:

15. 1. Find the volume of the frustum of a right
circular cone whose slant height is 10 cm and whose
radii are 3 cm and 9 cm.
Solution:
See we are given that r = 3, R = 9, and l = 10.
From the figure below, we see that the altitude
is
Hence, by Equation 4, we obtain
=
=
V π(8)(9 + 81 = 27)
=
312 cm3
2. The volume of a frustum of a right circular cone is
1176π cu. cm. The altitude of the frustum of a cone
is 18 cm. find the radii of the upper and lower
base if the product of their radii is 60 sq. cm.

17. 3. Find the volume and surface area of a frustum of a
cone having radius of the upper base equal to 4 cm
and radius of lower base equal to 6 cm, if it has a
height of 8 cm.
UNIT III
PRISMATOID
A
prismatoid
is
a
polyhedron
having
for bases two polygons in parallel planes, and for lateral

18. faces triangles or trapezoids with one side lying in one
base, and the opposite vertex or side lying in other base
of the polyhedron.
Properties:
• The altitude of a prismatoid is the perpendicular
distance between the planes of the bases.
• The mid-section of a prismatoid is the section
parallel to the bases and midway between them.
The volume of a prismatoid equals the product of one-
sixth the sum of the upper base, the lower base, and four
times the mid-section by the altitude.

19. EXAMPLE:
1. A trapezoidal canal having a base 6 m wide and 8 m
wide at the top at one end and a base width of 6 m
wide and 10 cm width at the top at the other end of
the canal which is 50 m long. Find the volume of

20. the earth excavated for the canal. The depth of the
canal is 4 m depth at one end and 5 m depth at the
other end.
2. Find the volume of the prismatoid shown.

22. ABOUT THE AUTHOR
Meryl Mae Rabut
Nelmida is the present
Vice-President of the
Louisian Mathematics
Society in Saint Louis
College (City of San
Fernando, La Union). She
shares her unique
intelligence in Mathematics
through the club’s program
such as remedial and
tutorials in Lingsat
Community School and Poro-
San Agustin Elementary School. She finished her Basic
Education in Christ the King College, City of San
Fernando, La Union. She is an active member of the Pantas
Circle during her high school years. The Pantas Circle
provides opportunities for students to hone their
knowledge in Mathematics. She is presently in her second
year of studying Bachelor of Secondary Education Major in
Mathematics.

23. UNIT I
Frustum of a Pyramid
UNIT II
Frustum of a Right Circular Cone
UNIT III
Prismatoid
UNIT IV
Truncated Prism