solid mensuration (solids with volume equals mean BH)

*****Frustum of a Pyramid

Frustum of a Right Circular

Cone

Prismatoid

Truncated Prism

*****

Meryl Mae R. Nelmida

Solid Mensuration

UNIT I

Frustum of a Pyramid

UNIT II

Frustum of a Right Circular Cone

UNIT III

Prismatoid

UNIT IV

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.

Note: figure 2 represents the unfold of a frustum of a pyramid

Properties:

Fig. 1

Fig. 2

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.Altitude2.Slant height3.Lateral edge4.Lower base edge5.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,

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

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.

Eqn. 1

Eqn. 2

Fig. 4

Let H = LP + altitude of pyramid P-MNRQH = 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 MNRQV = volume of the frustum P-MNRQV1 = volume of the pyramid P-M’N’R’Q’V2 = volume of the pyramid P-MNRQ

Then

By the equation Volume = (B × h)/3

Substituting (2) and (3) in (1) and rearranging the terms, we get

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

Substituting (6) in (40 and simplifying, we get Equation 2 which is

s = l2

S L2

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:

b = 42 = 16B = 82 = 64H = 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 cmp1 = 24 * 4 = 96 cmP2 = 14 * 4 = 56 cm

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.

Properties: 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

Fig. 5

Eqn. 3

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,

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

Where:

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,

Eqn. 3.1

But for a right circular cone, b = πr2 and B = πR2. Substituting these values in the above equation, we get

Where:

V = volume of frustum

h = altitude of the frustum

r = upper base radius

R = lower base radius

EXAMPLE:

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:

=

Eqn. 4

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.

=

=

=

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 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.

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 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.

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.

UNIT I

Frustum of a Pyramid

UNIT II

Frustum of a Right Circular Cone

UNIT III

Prismatoid

UNIT IV

Truncated Prism