12 multiple-choice questions, progressively harder.
A prism has pentagonal bases. Which are its face and edge counts?
Solution
Correct answer: C
There are two bases and five joining faces. Two base boundaries have five edges each, and five further edges join matching vertices.
The counts are seven faces and fifteen edges.
A pyramid has a seven-sided base. How many edges and vertices does it have?
Correct answer: A
Seven edges surround the base and seven connect base vertices to the apex. There are seven base vertices and one apex.
This gives fourteen edges and eight vertices.
Use the labeled rectangular box. What is the relationship between complete lines AB and CR?
Correct answer: D
A plane containing both lines would contain the noncollinear points A, B, and C and hence would be the bottom plane.
R lies outside that plane. The two lines are noncoplanar, so they are skew.
Use the labeled rectangular box. In what complete line do the top plane PQRS and right plane BCRQ intersect?
Correct answer: B
The top and right planes share Q and R.
They are distinct planes, so their full intersection is line QR, continuing beyond the edge segment shown.
Use the labeled rectangular box. Which complete line is parallel to the bottom plane ABCD?
PQ lies in the top plane, which is parallel to the bottom plane, and therefore has no point in the bottom plane.
AP, BQ, and CR meet the bottom plane at A, B, and C, respectively.
For the labeled cube net, which face is opposite A?
Hold B underneath. A and C fold upward on opposite sides of B.
They become the left and right cube faces, so C is opposite A.
Six equal squares form the two-by-three rectangle shown. All internal grid segments remain joined during folding. Why can it not be a cube net?
A, B, D, and E have the same joined corner O. Folding the shared edges cannot separate that corner.
Only three square faces meet at each cube vertex, so the four joined corners cannot fit there.
A box has front view 11 cm wide and 4 cm high, and top view 11 cm wide and 6 cm deep. What are its width, depth, and height?
The front fixes width at 11 cm and height at 4 cm. The top repeats the width and supplies depth 6 cm.
The right view would therefore have depth 6 cm and height 4 cm.
A front view of a rectangular box has height 5 cm. A proposed right view of that same box has height 7 cm. Both labels are exact. What follows?
Front and right orthographic views record the same vertical height.
Exact heights of 5 cm and 7 cm cannot describe the same box in the same fixed orientation.
A cylinder has congruent circular bases in parallel planes, and its center-to-center axis is not perpendicular to them. What is it?
Its two circular bases identify a cylinder.
The stated nonperpendicular axis makes it oblique; a cone would have an apex instead of a second congruent base.
A cube net is cut so two faces no longer share an edge on paper. What can happen after folding?
Some edges are separated by cuts only to let the surface lie flat.
When the cube closes, such cut edges can meet again, so separation in the net does not force separation on the solid.
A filled-column solid has two columns behind the left position of its front view. Their heights are 1 unit and 4 units. What front silhouette height appears at that position?
Both columns project onto the same width position. The taller one reaches height four and determines the silhouette.
The heights are not added because the columns are behind one another, rather than stacked.
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