Exercises: 3D space

In the world of video game design, objects are defined by their dimensions. State how many dimensions ($0$, $1$, $2$ or $3$) the following items that we find in a room have:

1. a) The cable of a pair of headphones stretched out.

1. b) The flat screen of a monitor (not counting its thickness).

1. c) A box where you keep your console games.

1. d) A point of light on the ceiling.

Look at the edges of this cube and determine the relative position (parallel, intersecting, or skew) of the following pairs of lines containing the edges:

1. a) The line passing through points $A$ and $B$ with respect to the line passing through $C$ and $D$.
2. b) The line passing through $A$ and $B$ with respect to the line passing through $B$ and $C$.
3. c) The line passing through $A$ and $B$ with respect to the line passing through $G$ and $H$.

Imagine that the walls, floor, and ceiling of your classroom are planes. State the relative position of:

1. a) The floor plane and the ceiling plane.
2. b) The plane of the whiteboard wall and the plane of the window wall (which is next to it).
3. c) A line formed by the corner between two walls and the floor plane.

We know that given two points in space, exactly one line passes through them.

1. a) How many lines can pass through a single point in space?
2. b) How many planes can contain a single line?
3. c) What is the minimum number of non-collinear points needed to define a single plane?

In space, there are lines that are not parallel but never intersect. These lines are called skew lines. Look at the drawing of the skate ramp and state their relative positions:

1. a) The top edge of the ramp ($r$) and the base line on the ground ($s$).
2. b) The vertical edge of the handrail ($t$) and line $s$.

Draw in your exercise book (or describe) the position of a line and a plane in the following cases:

1. a) The line is parallel to the plane (think of a fluorescent light on the ceiling and the floor).
2. b) The line is contained in the plane (think of a line painted on the floor).
3. c) The line intersects the plane (think of a pencil going through a sheet of paper).

You can do it on paper and send a photo with your answer.

Many geometric solids are generated by rotating a plane figure around an axis. These are called solids of revolution. In the drawing you see three objects with their axis of rotation. Identify which figure (rectangle, right triangle or semicircle) generated each of them when rotated around that axis:

1. a) A cylindrical soda can.
2. b) A traffic cone.
3. c) A pétanque ball.

Geometric solids whose faces are all polygons are called polyhedra. In any simple polyhedron, a famous relationship called Euler's formula holds: $Faces + Vertices = Edges + 2$.

Check whether this formula holds for a cube, knowing that it has $6$ faces, $8$ vertices, and $12$ edges. Write down the mathematical calculation.

Aoife is observing the structure of a metal shelf in her classroom and claims the following: “The front-left vertical bar and the back horizontal bar of the top shelf do not touch anywhere; therefore, they must necessarily be parallel lines, just like the lines I draw in my notebook.” Identify and explain the reasoning error that Aoife makes, name the actual relative position of these two bars in three-dimensional space, and clarify why her deduction would have been correct on a two-dimensional sheet of paper.

A sports hall has a roof shaped like a regular hexagonal prism, whose top and bottom bases are hexagons and its lateral faces are rectangles.

1. a) Before counting edges or applying any formula, reasonably estimate at first glance whether this solid will have more edges than the sum of its faces and vertices combined, or if the opposite will happen.
2. b) Count in an orderly way the number of faces ($C$), vertices ($V$), and edges ($A$) of the prism and check whether your initial estimate was correct using Euler's formula ($C + V = A + 2$).
3. c) Justify why the total number of edges obtained is reasonable, taking into account how many edges each base contributes and how many vertical edges connect both bases.

Consider a cube or regular hexahedron with bottom base $ABCD$ and top base $EFGH$, where the vertical edges are $AE$, $BF$, $CG$, and $DH$. Identify the relative positions of the indicated edges:

1. a) Edge $AB$ relative to edge $EF$.
2. b) Edge $AD$ relative to edge $CG$.
3. c) Edge $GH$ relative to edge $AB$.

In designing an advertising marquee shaped like a right triangular prism, we want to analyse the relationship between its planes. If we consider plane $\pi$ containing the lower base of the prism:

1. a) State which faces of the prism generate planes that intersect plane $\pi$.
2. b) State which edges of the prism are parallel to plane $\pi$.
3. c) What 2D shape is formed at the intersection of a lateral face with the plane of the base?

An architect is designing an avant-garde monument that is a convex polyhedron. To verify its structure, they need to check Euler's formula. The monument has $8$ hexagonal faces and $6$ square faces.

1. a) Calculate the total number of edges ($A$) of the polyhedron.
2. b) Using Euler's formula, determine the number of vertices ($V$) that the monument will have.

Imagine you have a flat metal piece in the shape of a right-angled trapezium, as shown in the diagram. If we rotate this shape through $360^{\circ}$ around the side measuring $8$ cm (the vertical axis):

1. a) What solid of revolution is generated?
2. b) Draw a sketch of the resulting solid, indicating its basic elements (radii and height).

You can do it on paper and send a photo with your answer.

In a room with perpendicular walls, consider one of the top corners of the ceiling as the origin of points. If we draw an imaginary line across the room from that corner to the opposite corner of the floor (diagonal of the room):

1. a) What relative position does this line have with respect to the floor plane?
2. b) Does this line intersect the line formed by the junction of two side walls? Justify your answer.
3. c) How many points in common does the diagonal have with the plane of one of the walls?

A cut is made through a right cylinder of radius $r = 4$ cm and height $h = 10$ cm using a plane. Describe what 2D shape is obtained if:

1. a) The plane is parallel to the bases of the cylinder.
2. b) The plane is perpendicular to the bases and passes through their centres.
3. c) The plane is oblique to the bases and does not intersect them.

Given a regular square pyramid, we label its base edges as $b_1, b_2, b_3, b_4$ and its lateral edges as $a_1, a_2, a_3, a_4$. State whether the following statements are true or false, giving reasons:

1. a) Any pair of lateral edges are skew lines.
2. b) A base edge and the lateral edge originating from one of its endpoints are intersecting lines.
3. c) Two opposite base edges are parallel to each other.

Look at the net of this geometric solid. Without performing calculations, answer:

1. a) What polyhedron will be obtained when folding it? Classify it completely.
2. b) How many dihedral angles will the solid have once assembled?
3. c) What is the order of each of its vertices (how many faces meet at them)?

At the AirNav Ireland air traffic control centre in Dublin, the trajectories of two aircraft flying over a flat area are monitored, whose ground we consider as a horizontal plane $\pi$. The trajectories of both planes are straight lines:

• Flight A flies stabilized at a constant altitude of $9\,500\text{ m}$ following a fixed heading to the north.

• Flight B performs a continuous climbing maneuver from $3\,000\text{ m}$ up to $11\,000\text{ m}$ heading east, crossing that same geographical sector.

1. a) Model the trajectory of Flight A as a line $r$. What is the relative position of line $r$ with respect to the ground plane $\pi$? Justify your answer.
2. b) When projecting both routes onto the flat radar screen (two dimensions), the tracks intersect at a common point on the screen. However, directly above that point Flight B passes at an altitude of $7\,000\text{ m}$, so the planes never collide in the air. Model both routes as two lines $r$ and $s$ in space and determine their actual relative position, justifying whether or not there exists a common plane that can contain both at the same time.

A craft design workshop receives an order to manufacture a closed polyhedron for a lamp. The client requires the structure to have exactly $15$ edges ($A = 15$) and to be a classical polyhedron (a right prism or a regular pyramid).

1. a) Applying Euler's formula ($C + V = A + 2$), write the numerical equality and determine what the exact value of the sum of the number of faces and vertices ($C + V$) must be.
2. b) If the structure were a pyramid whose base has $n$ sides, reason how many edges it would have in total in terms of $n$. Based on that deduction, explain why it is mathematically impossible to build a pyramid that fulfills the order of having exactly $15$ edges.
3. c) Set up the same deduction for a prism whose base has $n$ sides. Determine what type of prism fulfills the conditions, indicating how many faces and how many vertices the lamp will finally have.

Imagine you are designing a structure for an adventure park. You have a giant cube whose vertices are labelled as $A, B, C, D$ (lower base) and $E, F, G, H$ (upper base, such that $E$ is above $A$, $F$ above $B$, etc.).

If we draw a straight line passing through the midpoints of the edges $AE$ and $CG$, and another straight line connecting the vertices $B$ and $H$:

1. a) State the relative position of these two lines in space. Justify whether they intersect, are parallel, or are skew.
2. b) If we draw a plane passing through points $B$, $D$, and $H$, what type of cross-section (2D shape) is formed when cutting the cube?
3. c) How many edges of the cube intersect the plane defined by face $ADHE$?

An archaeologist has found a piece of crystal in the shape of a perfect regular octahedron. To protect it, they decide to build a wooden box in the shape of a cube that fits the vertices of the octahedron exactly.

1. a) If each edge of the octahedron measures $10 \text{ cm}$, calculate the number of faces, vertices, and edges of the shape and check if Euler's formula holds true.
2. b) Explain with reasons the relationship between the centres of the faces of the wooden cube and the vertices of the octahedron.
3. c) If we cut the octahedron with a plane parallel to one of its faces that passes through the exact centre of the solid, what polygon do we get in the cross-section?

In a design and technology class, students are building a scale model of a modern building. The central piece is a right regular hexagonal prism. To give it an artistic touch, they want to make plane cuts with an electric saw.

1. a) If the cutting plane passes through three non-consecutive vertices of the top base, describe the shape of the resulting cross-section.
2. b) Is it possible to make a cut such that the resulting cross-section is a rectangle? What about a square? Describe how the plane should be in each case.
3. c) If the prism has $12$ base edges and $6$ lateral edges, how many space diagonals (diagonals that do not lie on any face) does this polyhedron have in total?

A gift box manufacturer wants to create a special "dual" model. First, they make a regular tetrahedron with edge length $L$. Then, they join the midpoints of all the edges of the tetrahedron to each other.

1. a) What polyhedron is formed inside by joining these midpoints?
2. b) Calculate the number of faces, vertices, and edges of the resulting new polyhedron.
3. c) If the original tetrahedron satisfies Euler's formula ($C + V = A + 2$), show whether the interior polyhedron also satisfies it.

An interior designer wants to install a strand of LED lights across a rectangular room that is $6 \text{ m}$ long, $4 \text{ m}$ wide, and $3 \text{ m}$ high. The cable must go from a corner on the floor ($A$) to the opposite corner on the ceiling ($G$).

1. a) Calculate the minimum length of the cable if it is stretched taut through the interior of the room (the diagonal of the cuboid).
2. b) If the designer decides to run the cable along the walls (without going through the air), what would be the shortest distance between those same two points? (Hint: think about the net of the room).
3. c) State how many edges of the room are parallel to the line containing the height edge where point $A$ is located.

Consider a cheese cube with an edge length of $4 \text{ cm}$. We want to slice it with a very thin knife such that the cross section is a regular hexagon.

1. a) Describe precisely through which points on the edges of the cube the plane must pass to obtain this hexagon.
2. b) How many face diagonals does the cube have in total? And how many space (interior) diagonals?
3. c) If we call the plane containing the hexagon $P$, is there any edge of the cube that is parallel to this plane, or are they all intersecting? Justify your answer.

A logistics company uses containers shaped like a right trapezoidal prism (the base is an isosceles trapezium).

1. a) Determine the number of right dihedral angles this solid has.
2. b) If we extend the two non-parallel side faces, explain what would happen: Would they intersect in a line? Would they be parallel? What geometric solid would be formed if we cut that new structure with a plane parallel to the base?
3. c) Check whether a trapezoidal prism with bases of $5$ sides (if it were an irregular pentagonal prism) always satisfies Euler's formula.

We are studying relative positions in space within a complex polyhedron: a regular dodecahedron (12 pentagonal faces).

1. a) Choose any face as plane $\pi$. How many faces of the dodecahedron are parallel to $\pi$?
2. b) How many edges of the dodecahedron are parallel to a given edge?
3. c) If 3 faces meet at each vertex of the dodecahedron, and each face has 5 vertices, work out the total number of vertices of the solid (bearing in mind that each vertex is shared).

In an exhibition hall shaped like a cube with lower vertices $A, B, C, D$ (floor) and upper vertices $E, F, G, H$ (ceiling, located directly above $A, B, C$, and $D$ respectively), two straight tension cables are installed:

• Cable 1: connects vertex $A$ to the opposite upper vertex $G$ (main diagonal of the cube).

• Cable 2: connects the midpoint of the vertical edge $BF$ to the midpoint of the vertical edge $DH$.

To find out whether both cables touch each other at the centre of the room or cross at different heights, two students propose different approaches:

• Cian's method: Check if there is a diagonal plane containing the four endpoints ($A, G$, and the midpoints of $BF$ and $DH$). If such a plane exists and the lines are not parallel, then they intersect.

• Aoife's method: Determine the position of the midpoint of Cable 1 and Cable 2 with respect to the floor and the walls, checking whether both segments share exactly the same central point in three-dimensional space.

1. a) Compare both methods: explain which of the two seems clearer and more direct to you to justify the result without making visual perception errors.
2. b) Apply the method you considered best to conclude with reasoning whether the two cables are intersecting (they intersect at a point) or skew in space.

The Sheehan Towers in Dublin are a pair of prism-shaped buildings whose lateral faces are inclined with respect to the horizontal ground (plane $\pi$). Consider one of the inclined facades of the tower, which we call plane $\alpha$.

1. a) Determine the relative position between the plane of the facade $\alpha$ and the ground plane $\pi$. If we were to extend both planes indefinitely in space, what would be the resulting geometric figure of their intersection?
2. b) In a polyhedral model of the tower starting from a closed quadrangular prism ($6$ faces, $8$ vertices, and $12$ edges), a model maker decides to truncate (cut off) one of the top corners using a flat cut that passes close to a vertex without touching any of the other seven original vertices, removing that corner and creating a new flat triangular face in its place.

Calculate, with reasoning, the number of faces ($C'$), vertices ($V'$), and edges ($A'$) that the model will have after making this cut.

1. c) Check whether the new modified solid still satisfies Euler's formula ($C' + V' = A' + 2$) and mathematically justify why cutting off a corner in this way does not alter the balance of the formula.