12 multiple-choice questions, progressively harder.
A right box has a 3 cm by 4 cm base and height 6 cm. A plane perpendicular to the base contains a base diagonal. What are the true section dimensions?
Solution
Correct answer: C
The plane rises above the diagonal, making a rectangle. The diagonal length d satisfies
The upright dimension is the full 6 cm height.
A cube is sliced by a plane through the midpoints of three edges that meet at one vertex. The plane cuts off that corner. What is the section?
Correct answer: A
The plane meets the three faces adjoining that corner. Each face contributes the segment joining two of the given midpoints. Those three segments bound a triangle.
A right square pyramid is cut by the plane through its apex and a base diagonal. Which section results?
Correct answer: D
The base diagonal and the two equal apex-to-opposite-vertex edges bound the section. This plane includes the apex, so it gives a triangle.
Compare interior base-parallel sections of a prism and a pyramid while moving toward their tops. Which description is correct?
Correct answer: B
A prism repeats the same base shape by translation. A pyramid narrows to its apex, so its base-parallel sections become smaller.
An oblique plane cuts a right circular cylinder. Its complete section boundary lies on the curved side, and the plane misses both end disks. What is its true section?
The plane is tilted relative to the bases, so the complete closed side cut stretches in one direction. The true section is a noncircular ellipse, not just a perspective view of a circle.
Which selected cut makes a noncircular elliptical section of a right circular cone?
A base-parallel cut is circular, while an axial cut is triangular. The specified tilted plane makes a closed oval entirely on the curved side.
A sphere has radius 13 cm. A cutting plane is 5 cm from its center. What is the radius of the circular intersection?
The sphere radius is the hypotenuse of the right triangle formed with the perpendicular center distance and the section radius r.
A right cylinder has radius 5 cm. Find the exact area of an interior section parallel to its bases.
The true section is a disk of radius 5 cm. Square the radius for its area.
Rectangle ABCD has AB of 5 cm and perpendicular AD of 9 cm. Compare a turn about AB with a turn about AD. Which description is correct?
The axis side becomes the height. The other side becomes the radius. Thus the two named axes exchange the roles of 5 cm and 9 cm.
Right triangle ABC has right angle A, AB of 7 cm, and AC of 24 cm. The filled triangle turns about AB. What is the diameter of the cone base?
AB is the height-axis. AC sweeps the base radius, which is 24 cm, so the diameter is
A filled semicircular region has diameter 14 cm. It turns about that diameter. What is the radius of the ball formed?
The rotation preserves distance from the semicircle center. Its radius is half its diameter.
A right cylinder has height 10 cm and an axial section 6 cm wide. Which filled rectangle and axis generate it?
The axial width is a diameter, so the radius is 3 cm. The generator must have that 3 cm reach from a 10 cm axis side.
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