12 multiple-choice questions, progressively harder.
A right cylinder has an interior base-parallel section of area 49π49\pi49π square cm and height 11 cm. Which filled rectangle generates it by turning about one side?
Solution
Correct answer: D
The disk area determines the positive radius.
The fixed 11 cm side provides the height, and the perpendicular 7 cm side sweeps the radius.
The section through the axis of a right cone is an isosceles triangle with base 18 cm and perpendicular height 12 cm. Which filled region and axis generate the cone?
Correct answer: B
The axial base spans the whole diameter, so the cone radius is 9 cm. Half the axial triangle is a right triangle with radius-leg 9 cm and axis-leg 12 cm.
Every interior section of a pyramid parallel to its base is a pentagon. What does this determine about its base?
Correct answer: C
A base-parallel section has the same number of sides as the base. The pentagonal shape fixes five sides, but does not force equal side lengths or angles.
A plane cuts exactly six faces of a convex polyhedron, contains no polyhedron edge, passes through no vertex, and meets each of those faces in a boundary segment. What is the section?
Correct answer: A
The section boundary consists of the six segments supplied by the six crossed flat faces. A six-sided polygon is a hexagon.
A sphere is tilted on a drawing, and its central plane section is drawn as an oval. Which statement is forced by the geometry?
Every point of the central section lies in the plane and is one sphere radius from the sphere center. That is a circle, whatever the projection looks like.
Rectangle ABCD has AB of 6 cm and AD of 10 cm. A full turn about a side must produce a cylinder with height less than its radius. Which side works?
About AB, the height is 6 cm and the radius is 10 cm, satisfying the inequality. About AD, the height is 10 cm and radius 6 cm, which reverses the comparison.
A right triangle has legs 8 cm and 15 cm. Its filled region must generate a cone with base circumference 16π16\pi16π cm. Which leg must be the axis?
The circumference sets the radius.
The 8 cm leg must therefore sweep the radius. The perpendicular 15 cm leg is the axis.
A sphere has radius 10 cm. A plane lies 6 cm from the center. Find the exact area of the disk cut from the ball.
Use the sphere radius as the hypotenuse, then use the resulting squared section radius directly in the disk area.
Two parallel planes cut the interior of the same sphere. Each is 4 cm from its center, on opposite sides of the center. Compare the circular intersections.
Each section radius belongs to a right triangle with the same sphere-radius hypotenuse and the same perpendicular center distance. Their positive radii are therefore equal.
A filled right triangle with legs 6 cm and 636\sqrt363 cm turns about the 636\sqrt363 cm leg. What is the shape of the whole axial section of the cone?
The cone radius is 6 cm, so the whole axial base is 12 cm. Each sloping side has the length of the generating hypotenuse c.
All three sides of the axial section are 12 cm.
A cylinder is generated by turning a filled rectangle about a side. Its axial section is a square of side 14 cm. What rectangle and axis work?
A square axial section means the cylinder height and diameter are both 14 cm. The radius is 7 cm, so the generator reaches 7 cm from its fixed 14 cm side.
Compare turning only a semicircular arc with turning the filled semicircular region of the same radius about their common diameter. Which description is correct?
The boundary arc supplies only points on the sphere surface. The filled region includes the segments from the axis to that arc, so it fills the ball as it turns.
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