12 multiple-choice questions, progressively harder.
A prism has 18 vertices. What are its base type and face count?
Solution
Correct answer: D
The two congruent bases have the same number of vertices, so each has nine.
A nine-sided polygon is a nonagon. There is one joining face per base edge, in addition to the two bases.
There are eleven faces.
A pyramid has 10 edges. How many faces and vertices does it have?
Correct answer: B
A pyramid has as many apex-joining edges as base edges, so its ten edges split into two groups of five.
It has a pentagonal base, five triangular faces, five base vertices, and one apex.
For the labeled cube net, which set of three faces can meet at one cube vertex?
Correct answer: C
The opposite pairs are A with C, B with D, and E with F. A vertex cannot contain an opposite pair.
A, D, and E become left, top, and back and do meet at the top-back-left corner. Each other set contains an opposite pair.
A closed polyhedron is described as one pentagon and five triangles. Each triangle joins one pentagon edge to a common apex outside its plane. Which edge count follows?
Correct answer: A
The description is a pentagonal pyramid. Five edges surround the pentagon, and five edges connect its vertices to the apex.
Shared edges are counted once, so there are ten.
In the labeled box, what justifies that complete lines AB and QR are skew?
A, B, and Q are noncollinear front-face vertices and determine that plane.
If one plane contained both complete lines, it would contain these three points and R. Since R is outside the front plane, no such plane exists.
A top view has all six sites of a three-by-two unit grid occupied. A height plan gives back row 2, 1, 1 and front row 3, 2, 1. What front silhouette heights appear from left to right?
At each width position compare the back and front column heights. The larger height determines the silhouette.
The pairs are 2 with 3, 1 with 2, and 1 with 1, giving 3, 2, and 1.
A filled-column solid has front silhouette heights 3, 2 from left to right, right silhouette heights 3, 2 from front to back, and a full two-by-two top footprint. Which height plan fits? Rows are listed back first, then front.
For back 2, 1 and front 3, 2, the width-position maxima are 3 and 2; the depth-position maxima are 3 at the front and 2 at the back. Every site is occupied.
The other plans respectively give back height 3, front-right height 3, or an empty top-view site.
Only outer silhouettes are specified: front and right are two-by-two squares, and the top is a full two-by-two square. Filled columns have positive integer heights. Which conclusion is justified?
Both arrangements occupy every top site. In the diagonal plan, each width position and each depth position has a column of height two.
Therefore both give the specified silhouettes, even though the full block has no shorter column.
A right triangular prism has triangular bases with side lengths 5 cm, 6 cm, and 7 cm. Its joining edges are 9 cm long. Which surface pieces belong to its net?
Each triangular base appears once. Each of the triangle's three sides is joined through the 9 cm prism length, making one rectangle.
Thus the three rectangles match the three base side lengths, each paired with 9 cm. No area calculation is needed.
A rectangular box has a 10 cm by 6 cm front view. No top or side view is supplied. Which additional measurement determines the third dimension?
The front view already determines width and height. Its diagonal, perimeter, and bottom edge all belong to that same front plane.
The missing independent dimension is front-to-back depth; a perspective impression cannot supply it.
In the labeled rectangular box, which statement about complete planes is correct?
The adjacent rectangular-box face planes meet at a right angle. They share B and Q, so their full intersection is line BQ.
Sharing a line does not make two distinct planes coincide, and planes extend beyond face boundaries.
A prism has two congruent parallel triangular bases. Its joining edges are oblique to the bases. Which statement is guaranteed?
Tilting the joining edges changes angles, not which faces, edges, and vertices are joined.
There remain two triangular bases, three joining parallelograms, six base edges, three joining edges, and six vertices.
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