Study Guide

The Third Dimension

The Third Dimension

Up until now, we've explored the world in two dimensions. Area, shapes, and x-y plots are all good and fine, but they're a little flat, aren't they? After all, you can't really play Minecraft in two dimensions. If we crank geometry into third gear, we'll go from kiddy cartoons to sophisticated computer animation, like Paper Mario plopped into a 3D Mushroom Kingdom.

The first step in three-dimensional space is knowing how to look at it. Yes, with your eyes, smarty-pants, but it's more than just that. We may think we see the world in three dimensions, but we really only see it in two. What a bust. That means we need to come up with ways to draw 3D space on a piece of 2D paper.

  • Drawings

    One way to represent 3D figures is with an orthogonal drawing, which is a fancy word for drawing all the different sides of the three-dimensional shape. That means a cube would look like this:

    It's useful, because we can figure out the actual 3D shape if we have the orthogonal drawings. But if you ask us, it's a little repetitive and not very interesting. We signed up for 3D, and by golly, we're going to get 3D.

    Our eyes see shapes with perspective (3D glasses optional), so what better way to draw these shapes than through corner or perspective drawings?

    Telling you that the picture above is a cube would be unnecessary. You know it's a cube. You've seen it a million times in your freezer's ice tray. But that's exactly the point: to draw shapes the way we see them.

    So what if we had orthogonal drawings and we wanted to find out the perspective drawing of the shape? Let's say we have this orthogonal drawing:

    The thin lines on the front and top indicate some kind of change in the surface when we look at the image from those viewpoints. The side view doesn't have any change. If we look at the images and piece them together like Legos, we should get a perspective drawing that looks like this:

    With a little practice, you'll be able to translate orthogonal drawings to perspective drawings faster than you can say, "translate orthogonal drawings to perspective drawings."

    So what kinds of things are three-dimensional shapes? Well, lots! Anything that exists in the real world is a three-dimensional shape. Like your computer. And your eyeball. And your little sister's Furby that you're convinced is going to kill you in your sleep.

    Terrifying, isn't it?

    But before we go and classify Furbies as anything at all, we should start with the basics of the third dimension.

    If we have a solid with all flat surfaces that completely encloses a region of space, we call it a polyhedron. "Poly-" means many and "-hedron" means face.

    The line segments of intersection are called edges and the points are called vertices.

    A regular polyhedron has all regular congruent polygons for faces, and the edges are all congruent. There are only five types of regular polyhedrons, called the Platonic solids, named after the Ancient Greek philosopher, Plato, who also invented the plate (except not really).

    A prism is a special type of polyhedron with two parallel congruent faces called bases. The other sides are parallelograms. If a prism is regular, that means its bases are regular polygons.

    A pyramid has one base with all the other faces intersecting at one vertex. Some pyramids are named after a city or civilization (the Cairo Pyramids, the Mayan Pyramids, etc.), but we'll name our pyramids for their bases.

    Since the base for this pyramid is a square, it's a square pyramid.

    Some solids aren't polyhedrons. These are shapes that have faces that aren't flat like polygons. They're round or curved.

    A cylinder is a pair of congruent circular bases on two different parallel planes. A cone is what you eat your ice cream in, a circular base with a single vertex. A sphere is a ball, but its official definition is a set of points equidistant from a given point in 3D space. They're all drawn below.

    These perspective drawings of the shapes are their ID badges. If we see one, we should be able to identify it without any problems.

    Sample Problem

    What is the name of the shape? What are the faces, vertices, and edges?

    The faces are the sides: ∆ABC, ∆ACD, ∆ADB, and ∆BCD.

    The vertices are just A, B, C, and D.

    The edges are the line segments AB, AC, AD, BC, BD, and CD.

    Since all the faces are congruent equilateral triangles, this shape would be a regular polyhedron (one of the five Platonic solids). Since it has four faces total, it's called a tetrahedron. ("Tetra" means 4. And don't you tetra-get it.)

    Speaking of edges and faces and vertices, there's a helpful formula that applies to every polyhedron, no matter how crazy it looks. According to an 18th-century Swiss dude named Leonhard Euler, the number of faces plus the number of vertices in a polyhedron equals the number of edges plus 2. That's a mouthful, so here's the math version:

    F + V = E + 2

    Euler was the man, so this little trick goes by Euler's Formula.

    If you're still confused on 3D shapes, check out our video right here to help you out:

  • Nets

    When we slice and dice 3D shapes, we end up with interesting 2D figures. What we're doing is intersecting a plane (a 747, to be exact) with the shape to find its cross-section. We'll turn on the fasten seat belt sign, as there may be some turbulence.

    If we intersect a horizontal plane with a cylinder, we get a circle. But if we use a vertical plane, we get a rectangle. Cross-sections are useful to understand if you ever want to cut things into pieces. Like papers or muffins or that Furby that's still giving you nightmares. Yikes.

    If we can make 2D shapes out of 3D ones, we should be able to make 3D shapes out of 2D ones. We can! It's called origami.

    For those of us less skilled in the Japanese art of paper-folding, we can use nets, which are essentially the same thing. Nets look like cut open cardboard boxes, like this:

    Nets are patterns for creating 3D shapes. There can be multiple nets for the same shape, but they can't have any overlapping sides. For instance, the net for a square pyramid can look like either one of these:

    The reason nets are useful is because they catch fish and calculate surface area. Surface area is the sum of the areas of each face on a solid. It's the same as calculating how much wrapping paper you'd need to cover the entire solid perfectly. That'll definitely come in handy when you "accidentally" break your sister's demonic Furby and your parents make you buy her a present to make up for it. Actually, you might as well start working on surface area now.

    Sample Problem

    What's the surface area of this triangular prism?

    If we draw the net for this figure, it should look like this. The sides are labeled with the units we know and the areas are A through E.

    The sum of all the areas of the triangles and rectangles should give us the surface area. That means SA = A + B + C + D + E. What are we learning here, math or the alphabet?

    Now we can get down to calculating the areas of these puppies. C is a square with side length 6, so its area is:

    C = s2
    C = 62
    C = 36 units2

    We know the dimensions of D, so we can calculate the area of that rectangle.

    D = bh
    D = 6(10)
    D = 60 units2

    So far, so good. To calculate A, B, and E, though, we need to find the height of the prism. We can use the Pythagorean Theorem to do that.

    a2 + b2 = c2
    62 + b2 = 102
    b2 = 64
    b = 8 units

    That wasn't too bad. Now to find the rest of the areas. First the rectangle.

    B = bh
    = 6(8)
    B = 48 units2

    Now the triangles.

    A = ½bh
    A = ½(6)(8)
    A = ½(48)
    A = 24 units2

    Since A = E, the area of E is also 24. We're ready to plug in our values and find the surface area.

    SA = A + B + C + D + E
    SA = 24 + 48 + 36 + 60 + 24
    SA = 192 units2

    Get out your chainsaw, find all the cross-sections of that Furby, and then gift-wrap something way less creepy for your sister.