Solids, Nets, and Spatial Relationships
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Learning goals
- Classify solids by their bases, faces, edges, vertices, and curved surfaces
- Distinguish right and oblique solids using their relationship to the base planes
- Justify line and plane relationships, including skew lines, on a labeled solid
- Test a proposed net by tracking faces and corners during folding
- Construct compatible solids from named orthographic views and explain possible ambiguity
Surfaces tell us what kind of solid we have
A solid is a three-dimensional object or region. Its boundary is its surface. A polyhedron has a closed surface made entirely of flat polygonal faces. Where two faces meet along a segment, that segment is an edge. The corners where edges meet are vertices (one corner is a vertex).
A triangular prism has two triangular ends and three joining faces. Counting the complete object, including hidden parts, gives five faces. Its two triangles contribute six vertices and six boundary edges; three more edges join matching vertices, giving nine edges.
A prism has two congruent polygonal bases in parallel planes, joined at corresponding vertices by parallel edges. Its other faces are parallelograms. The bases need not be the faces currently resting on a table. A pyramid has one polygonal base and an apex outside its plane; one triangular face joins the apex to each base edge.
Worked example 1 Count a hexagonal prism
Problem. A prism has hexagonal bases. Find its numbers of faces, edges, and vertices. Explain how your count includes the hidden parts.
Solution. Each base has six corners. There are two separate bases, so there are twelve vertices.
There are six edges around each base and six joining edges.
The two bases and six joining faces give eight faces.
The far base and hidden edges still belong to the solid; visibility does not change the counts.
A cylinder has two congruent circular bases in parallel planes, with corresponding circle points joined by parallel straight segments to form its curved side. A cone has one circular base and an apex outside its plane; straight segments from the apex to the base circle form its curved side. These curved sides are not polygonal faces, so cylinders and cones are not polyhedra. We reserve the face, edge, and vertex counts above for polyhedra; a circular rim is not one of their straight edges.
A sphere is the surface consisting of all points at one fixed positive distance from a center. The surface together with its interior is a ball. Neither has polygonal faces. Everyday language sometimes calls the entire ball a sphere; we will name the distinction when it matters.
Check your understanding
Which description identifies a pyramid?
A pyramid has one polygonal base. Each base edge joins the same off-plane apex to make a triangular face. The other descriptions identify a prism, a cylinder, and a sphere.
Right and oblique describe geometry, not the drawing
For a right prism, the edges joining its bases are perpendicular to the base planes. For an oblique prism, those edges lean relative to the base planes. Both still have congruent parallel bases. A right cylinder has its center-to-center axis perpendicular to its bases; an oblique cylinder does not.
The right-angle condition comes from the stated solid or a geometric mark. A right prism can look slanted after you turn it or draw it in perspective. A page angle is not automatically the angle in space.
Check your understanding
A prism is turned so its joining edges look slanted on the page. They are still perpendicular to its base planes. How is it classified?
Right refers to the angle between the joining edges and the base planes. Turning the whole object changes its picture, not that relationship.
Lines and planes in space
Use complete lines when classifying relationships, even if a drawing shows only edge segments. Two distinct lines are parallel when they lie in one plane and never meet. They intersect when they share a point. Skew lines do not lie in one plane; they neither meet nor run parallel. Two segments missing each other on the page do not settle which relationship their complete lines have.
In this rectangular box, lines AB and PQ are parallel because they are opposite sides of the front rectangle. Lines AB and BQ are perpendicular because they meet at a right corner of that rectangle.
Distinct planes either have no common point and are parallel, or intersect in a complete line. The bottom and top face planes are parallel. The front and right face planes intersect in line BQ, continuing beyond the box. Adjacent face planes of a rectangular box meet at right angles and are perpendicular planes.
A line parallel to a plane has no point in that plane. Line PQ is parallel to the bottom plane. A line perpendicular to a plane meets it at a right angle to every line of the plane through the meeting point. In the box, line AP is perpendicular to the bottom plane at A.
Worked example 2 Justify a spatial relationship
Problem. Use the labeled rectangular box above.
Part A. Classify complete lines AB and CR, and justify your conclusion.
Part B. Identify the intersection of the front and right face planes.
Solution. Part A. A plane containing both lines would contain A, B, and C. These noncollinear points determine the bottom plane. But that plane does not contain R, which lies above it. No plane contains both lines, so AB and CR are skew.
Part B. The front and right face planes share B and Q. They are distinct planes, so their intersection is exactly line BQ. The shared edge is only the portion of that line on the box.
Check your understanding
In the labeled box, which relationship holds between complete line AP and the bottom plane ABCD?
AP meets the bottom plane at A and is perpendicular to that plane in a rectangular box. It does not lie in the bottom plane because P is outside it.
A net has to close without overlap
A net is a connected flat arrangement of a solid’s surface pieces that folds into the complete surface without gaps or overlapping interiors. In a polyhedron net, shared fold edges remain joined. Some edges cut apart in the net meet again when it closes. Faces that are separated on paper can therefore become neighbors.
Worked example 3 Follow faces while folding
Problem. In the net above, hold B as the bottom. Fold A, C, E, and F upward. Which face closes the top, and which face is opposite E?
Solution. A and C rise on the left and right of B. E and F rise at the back and front. D is attached beyond C, so a second quarter-turn brings D across the top. Thus D is opposite B.
E and F stand on opposite sides of the bottom, so F is opposite E. Their separation on the net agrees with their positions on the folded cube.
Before reading the next paragraph, predict whether the six-square rectangle below can make a cube while all the internal grid edges remain joined.
It cannot. At O, four square face corners are joined. Every square corner in a cube net must become a cube vertex, but exactly three faces meet at a cube vertex. Folding cannot separate those four corners while their grid edges remain joined. Having six equal squares is not enough; their arrangement matters.
Combine views without inventing hidden information
An orthographic view looks straight at an object along parallel sight lines, rather than using perspective. Fix the directions first. Here the front view looks from front toward back, the top view looks downward, and the right view looks from right toward left.
The front view records width and height. The top view records width and depth. The right view records depth and height. We place front at the left of the right view and back at its right. In the top view, back is above front. A dimension shared by two views must agree.
Views may include dashed hidden edges, but only when the drawing says they do. An outer silhouette alone does not supply every hidden edge or fill every space behind it.
Worked example 4 Check three views of a box
Problem. These views describe one rectangular box. Read its width, depth, and height, and explain the consistency checks. The panels are schematic; use the labels, not their drawn sizes. Unlabeled vertical numbers are in centimeters.
Solution. The width is 7 cm, appearing across the front and top. The depth is 5 cm, appearing in the top and right views. The height is 3 cm, appearing in the front and right views. Each repeated dimension agrees, so the views are compatible with a 7 cm by 5 cm by 3 cm rectangular box.
Check your understanding
A front view labels a box 8 cm wide and 4 cm high. Its top view labels it 8 cm wide and 6 cm deep. Which right view is compatible?
The right view must share depth with the top view and height with the front view. It must therefore show depth 6 cm and height 4 cm.
Worked example 5 Build two compatible solids
Problem. A solid consists of filled vertical columns of unit cubes on a two-by-two footprint, with no gaps inside any column. All four sites are occupied. Only the outer silhouettes in these views are specified; the dashed grid is a unit guide, not a set of hidden edges. Before reading the solution, sketch two different height plans that fit all three views.
Solution. A height plan puts one column height in each top-view square. One possibility uses height two at back-left and front-right, with height one at the other sites.
Another possibility has height two at every site.
In each plan, the tallest column at each left-to-right position has height two, so the front silhouette fits. The tallest column at each front-to-back position also has height two, so the right silhouette fits. All four sites are occupied, so the top footprint fits. The plans describe different solids: Plan A has two shorter columns. These silhouette views leave that information undecided.
Check your understanding
For a filled-column solid, what sets the front silhouette height at one left-to-right position?
Looking from front toward back projects all columns behind that width position onto the same vertical line. The tallest one sets the top of the silhouette; shorter ones may be hidden.