12 multiple-choice questions, progressively harder.
A pyramid has a trapezoidal base with parallel sides 777 cm and 131313 cm, separated perpendicularly by 444 cm within the base. Its perpendicular solid height is 999 cm. Find its volume.
Solution
Correct answer: C
The base area is the average of its parallel lengths times their separation.
This gives 404040 square centimeters. Use the 999 cm solid height.
The volume is 120120120 cubic centimeters.
A right circular cone has radius 555 cm and slant height 131313 cm. Find its exact volume.
Correct answer: A
Radius and perpendicular height are legs; slant height is the hypotenuse.
Use the recovered height.
A pyramid has base area 757575 square cm and perpendicular height 101010 cm. A base-parallel section is 444 cm above the base plane. What is its area?
Correct answer: D
The section is six centimeters below the apex, because its level is four centimeters above the base.
The area is 272727 square centimeters.
A pyramid has volume 919191 cubic cm and perpendicular height 777 cm. What is its base area?
Correct answer: B
Rearrange the pyramid-volume relation to solve for base area.
The base area is 393939 square centimeters.
A circular cone has volume 54π54\pi54π cubic cm and perpendicular height 181818 cm. Find its radius.
Solve first for the squared radius.
The positive radius is 333 cm.
A cone and a pyramid each have base area 606060 square units. The cone has perpendicular height 121212 units; the pyramid has height 999 units. What is the ratio of cone volume to pyramid volume?
Both volume expressions contain the same factor, one third of sixty. The ratio is therefore the ratio of the two perpendicular heights.
In simplest form this is 4:34:34:3.
An oblique circular cone has base radius 666 cm. Its apex is 777 cm perpendicularly from the base plane and 999 cm from the base center along a slanted segment. What is its exact volume?
The perpendicular height is seven centimeters. The slanted apex-to-center distance is not the height.
The volume is 84π84\pi84π cubic centimeters.
A triangular prism with base area 151515 square cm and perpendicular height 888 cm is cut into the three tetrahedra in the lesson. What is the volume of one piece?
The prism volume is base area times height.
The proved dissection has three equal-volume pieces, so each has volume 404040 cubic centimeters.
A cone has volume 96π96\pi96π cubic cm and radius 444 cm. What is its perpendicular height?
Substitute the radius and solve the resulting linear equation.
A pyramid has a square base, perpendicular height 121212 cm, and volume 144144144 cubic cm. Find the side length of its base.
Recover the base area before taking a square root.
The area is 363636 square centimeters, so the positive square side is 666 cm.
A circular cone has perpendicular height 121212 cm. A base-parallel section 444 cm below its apex has radius 333 cm. What is the whole cone volume?
The section-to-base length fraction is one third. Therefore the base radius is three times the section radius, or nine centimeters.
The volume is 324π324\pi324π cubic centimeters.
A cone and a pyramid have equal positive base areas B and equal perpendicular heights h. At distance z above their base planes, which expression gives the section area for both, throughout the height?
The distance down from the apex is h minus z. Divide that distance by h and square for area.
This equality holds at every corresponding height, which supports a Cavalieri comparison.
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