We live in a three-dimensional world. Every one of us has height, width, and length. Shapes exist in our 3D world, too: game dice, cuboids, donuts, pyramids, beach balls, traffic cones. All of those are 3D shapes. Let's take a closer look.

Table Of Contents

  1. Area and Volume
  2. The Mathematics of Three Dimensions
  3. Examples of 3D Shapes
  4. Mathematical Oddities

Area and Volume

Two-dimensional shapes, like polygons and circles, take up area, a space on a flat surface (like a plane). 2D shapes have width and length but no height.

Shapes in three dimensions (3D) have volume; they take up space and have width, length and height, thickness or depth.

You take up all three dimensions, and so do most of the objects around you. You must have space for everything with three dimensions to fit into the space you have. No two objects can be in the same space at the same time.

Suppose you are sitting in a chair, studying. Your dog leaps into your lap. Can your dog occupy exactly the same space you occupy? No, your dog may try, but the two of you cannot be in the same place at the same time. That is one way to know you and your dog are three-dimensional creatures.

The Mathematics of Three Dimensions

Shapes that exist in three dimensions all have width and length, just like their 2D relatives, but have an added dimension, height, which can also be called depth or thickness.

In mathematics, an infinite number of squares could occupy the same area on a plane, because they have no thickness or height. Adding the third dimension, this becomes impossible. A cube -- the 3D version of a square -- will have identical measurements in width, length and height. No other cube can occupy the space it takes.

[consider an animation showing unending series of squares leaving a mathematical plane like sheets of paper; next to it, a single cube that cannot be budged from its position by another cube bumping into it]

In real life, you cannot have squares on top of squares taking up no space. A sheet of notebook paper may seem very thin, but think about a package of 200 sheets -- it has measurable thickness. One sheet would be 1/200th the thickness of the whole stack, so even a sheet of paper is a 3D object.

Examples of 3D Shapes

3D solids are convex shapes having width, length and height (or depth, or thickness). They occupy space, meaning they have volume. You are familiar with many models of these 3D shapes:

  • Dice -- cubes
  • Shoe box -- cuboid or rectangular prism
  • Ice cream cone -- cone
  • Globe -- sphere
  • aperweight or Egyptian tomb -- pyramid
  • Soda can -- cylinder

Many other 3D solids exist, too. A whole category is called the Platonic Solids:

  1. Tetrahedron -- four triangular faces
  2. Cube -- six square faces
  3. Octahedron -- eight triangular faces
  4. Dodecahedron -- 12 pentagonal faces
  5. Icosahedron -- 20 triangular faces

[consider drawings of these five figures]

Mathematical Oddities

Unusual 3D shapes in mathematics include the torus, which looks like a donut:

[insert drawing: torus; try here for example: http://mathworld.wolfram.com/Torus.html]

Another strange shape is the pentagrammic prism, which is found in December ornaments and gift boxes:

[Insert drawing: pentagrammic prism; http://mathworld.wolfram.com/PentagrammicPrism.html]

Even our own planet earth is an unusual type of 3D shape, since it is not a true sphere. It is an oblate spheroid:

[Insert cartoon drawing of earth, based on https://earthhow.com/shape-of-the-earth/]

One of the most unusual 3D shapes, revealed only recently, is the scutoid. Once mathematicians and scientists "discovered" it, they found it already in use in insects and human skin cells!

[insert drawing of two connected scutoids, available here: https://www.nature.com/articles/s41467-018-05376-1#rightslink]

Next Lesson:

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Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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