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MATH 109 – SOLID MENSURATION
Solids for which V = (mean B) H
Frustum of a Cone
A frustum of a pyramid (cone) is a portion
of a pyramid (cone) included between the base
and the section parallel to the base not passing
through the vertex.
* Frustum of a Right Circular Cone
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. It is denoted by L .
Formulas
The lateral area of the frustum of a right circular cone is equal to one-half the sum of the
circumference of the bases multiplied by the slant height.
If LA is the lateral area, L the slant height, and R and
r the radii of the bases of the frustum of a right circular cone,
then
1
2
LA   2 r  2 R  L
1
2
LA   2  r  R  L
LA    r  R  L
The volume of a frustum of a right circular cone is equal to one-third the product of the altitude,
pi, and the sum of the square of the smaller base radius, square of the larger base radius and the
product of the two radii.
If V is the volume of a frustum of a right circular cone, H is the altitude and r and R the radii
of the bases, then
V


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A1  A2  A1 A2 H since A1   R2 and A2   r 2
3
V   R2   r 2   R2  r 2   H

3
1
V
1
 R2   r 2   Rr  H
3
V    r 2  R 2  rR  H
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Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone
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Problem 1. A reservoir contains 54,000,000 gal. of water when full. Find the depth of the water if the
reservoir is in the form of the frustum of a right circular cone of upper and lower base radii 200 ft. and
100 ft., respectively.
Problem 2. A 2 m. flagpole has an upper and lower diameter of 10 cm. and 40 cm., respectively. Find
the cost of painting at Php 100/ sq. m.
Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone
3
Problem 3. The height of a frustum of a cone is 1m.; the diameter of one of the base is twice that of the
other, and the slant height is inclined 45° with greater base. Find the volume and lateral area.
Problem 4. A manufacturer has an order for water pails (with cover), each in the form of a frustum of a
cone. Each pail has a slant height of 20 cm. If the diameters of the bases are 16 cm. and 12 cm.
respectively, how many of these pails can be made from 150,000 square centimeters of tin sheets,
given that 15% of the material will be wasted.
Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone
4
Problem 5. A container is in the shape of a frustum of a cone. The circumference of the bases of te
frustum are 120 cm. and 80 cm., and its height is 60 cm. Find the capacity of the container in liter.
Problem 6. How many cubic yards of earth will be required to fill 2,000 flower pots if the inside
diameters of the upper and lower bases are 15 inches and 8 inches, respectively and the depth is 15
inches.
Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone
5
Problem 7. A chimney in the form of a frustum is to be made of concrete 5 cm. thick. The inside
dimensions of the upper and lower bases of the chimney are 150 cm. and 250 cm., respectively. If the
height of the chimney is 300 cm., find the amount of material used in its construction. Find the weather
surface.
Problem 8. The inside diameters of the base of a water pail are 15 inches and 19 inches, and the slant
height is 21 inches. Find the capacity of the pail in pints.
Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone
6
Problem 9. The upper portion of the funnel is in the form of a frustum of a right circular cone whose
base diameters are 20 cm. and 5 cm., and whose height is 30 cm. The lower portion is in the form of a
right circular cylinder whose height is 10 cm. Find the area of the material necessary to make 1,000
such funnels if 10% of the materials are allowed for wastes and seams in each funnel.
Problem 10. A manufacturer has an order for 20,000 megaphones. The megaphones, conical in shape,
are to be 2 inches in diameter at the smaller end, 8 inches in diameter at the other end, and 1 ft. long. If
10% of the material used in manufacturing them will be wasted, how much material should be ordered?
Acorda, G., et al., (2005). Solid Mensuration (Workbook). Kumintang Ilaya, Batangas City: Feliber Publishing House
nd
Bland, J. R., & Kern W. F. (1938). Solid Mensuration. 2 Edition. John Wiley & Sons, Inc.
Solid Mensuration, http://www.math-principles.com
http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/frustum-right-circular-cone