041 Review Problem - Weight capacity of industrial soap kettle

Problem 41
Soap kettles used in the commercial manufacture of soap are as a rule large cylindrical vats, 50,000 lb. or more of soap being made in a single beating. Find the capacity of such a kettle having an inside diameter of 18 ft and an altitude of 30 ft. if soap weighs 70 lb. per cu. ft.

039 Review Problem - Bushels of wheat the grain elevator can hold

Problem 39
A grain elevator in the form of a frustum of a right circular cone is 24 ft. high, and the radii of its bases are 10 ft. and 5 ft., respectively; how many bushels of wheat will it hold if 1-1/4 cu. ft. equals 1 bu.?

037 Review Problem - Amount of material the factory chimney contain

Problem 37
A factory chimney is in the form of frustum of regular square pyramid. The chimney is 125 ft. high the edges of its bases are 12 ft. and 8 ft., respectively. The cross-section of the flue is 6 ft. square. How many cubic feet of material does the chimney contain?



032 Review Problem - How many cups of coffee a coffee pot can hold?

Problem 32
A coffee pot is 5 in. deep, 4-1/2 in. in diameter at the top, and 5-3/4 in. in diameter at the bottom. How many cups of coffee will it hold if 6 cups equal 1 quart? Answer to the nearest whole number.

025 Review Problem - Time required to fill a reservoir of water

Problem 25
A reservoir 10 ft. deep is in the form of a frustum of inverted square pyramid with bases of 100 and 90 ft. on a side respectively. How long will it require an inlet pipe to fill the reservoir if the water pours in at the rate of 200 gal. per min.? (One gal. = 231 cu. in.)

022 Review Problem - Tin required to create a funnel

Problem 22
How many square feet of tin are required to make a funnel, if the diameters of the top and bottom are 28 in. and 14 in., respectively, and the height is 24 in.?

013 Review Problem - Volume of water inside the Venturi meter

Problem 13
The accompanying figure represents the longitudinal view of a Venturi meter, a device designed to measure the flow of water in pipes. If the throat of the of the meter is 6 in. long and has an inside diameter of 4 in., find the volume of water in the meter which is used in 12-in. pipe line if the altitudes of the tapering parts are in the ratio 1:3 and the smaller altitude measures 12 in.




Frustum of a pyramid (or cone) is a portion of pyramid (or cone) included between the base and the section parallel to the base not passing through the vertex.

$V = \frac{1}{3}\left( A_1 + A_2 + \sqrt{A_1A_2} \right)h$


Frustum of a cone and frustum of a pyramid


Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

The lateral area of frustum of a right circular cone is given by the formula

$A = \pi (R + r) L$


R = radius of the lower base
r = radius of the upper base
L = length of lateral side

Derivation of formula for volume of a frustum of pyramid/cone

Frustum of a pyramid and frustum of a cone

Frustum of a pyramid and frustum of a cone


The formula for frustum of a pyramid or frustum of a cone is given by

$V = \dfrac{h}{3} \left[ \, A_1 + A_2 + \sqrt{A_1A_2} \, \right]$


h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum)
A1 = area of the lower base
A2 = area of the upper base
Note that A1 and A2 are parallel to each other.

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