1 Activities for this section:

Shaping up, The fab four, Solid shapes, Fighting crime

2 The purpose of definitions

Our goal in this chapter is to compare and contrast shapes. But before we can do that, we need to understand what we mean when we use different terminology for shapes. In other words, we need to have good definitions.

Definitions play a very important role in mathematics: they tell us specifically what we are working with. It can be hard to understand why this is important, but here are two examples that could possibly help.

First, here’s an example from our previous course. I could make a statement like “$8$ is a factor of $40$”, but to verify whether or not my statement is correct, you would first need to know what it means to be a “factor”. We used the definition of a factor in this instance: $A$ is a factor of $B$ if we can write $B = A \times N$ for some whole number $N$. So, in this case, you can say that $8$ is indeed a factor of $40$ because $40 = 8 \times 5$, and that $5$ is a whole number playing the role of $N$. At this point, we don’t have any questions or doubts about whether my statement is correct, because our definition was satisfied.

Second, here’s an example that we’ll expand on in a moment. I could make a statement like “the figure below is a square”.

But is it? It looks a little square-like, but that’s not enough for me to be certain. Am I saying that this object is definitely a square and I am bad at drawing? Am I trying to check whether this object is a square? I have lots of questions. We need a definition!

Children as young as preschool and kindergarten learn to identify shapes by their names, using their experiences and a collection of examples and non-examples that they remember. But as children progress through school, they should learn to be more and more specific about how and why they are classifying shapes, including learning to use precise definitions. As teachers, we want you to know and understand the definitions so that you can help children move towards using those definitions.

Good definitions are specific and actionable. When we use them, we know exactly what we are working with.

Good definitions are also easy to use. Since we want to use them a lot, we don’t want to include extra items in our definitions, because that would leave us with more things to check when we check the definition.

Good definitions are a habit in good mathematical work. Any time you are working with an object, its definition should be nearby!

3 Shapes

We are going to discuss together in class how and why we have chosen these definitions, and why they are good definitions. If you weren’t able to be in class for that session, it’s important for you to stop and look through the Shaping Up activity before you continue reading.

We hope you will use the rest of this section as a reference guide. Return to it frequently as you need these definitions!

In this section, we’ll discuss 2D shapes. Keep in mind, however, our discussion about dimension: when we talk about a 2D shape, in order for the shape to actually be two-dimensional, we need to include both the line or lines that draw the shape as well as the inside of that shape.

Notice that a vertex doesn’t have to be labeled with a dot in order to be a vertex.

Another way that some people like to think about a shape being closed is that it might hold water or that we have an inside and an outside of the shape.

We have special kinds of polygons that we will talk about frequently.

Often, it can be easier to write out the definition of a polygon and substitute the specific number of edges you are working with. For example, here is the definition of a triangle again.

We also have other special quadrilaterals whose names mostly come from using Greek prefixes for numbers. For instance, “hexa-” means $6$ in Greek, so a “hexagon” is a polygon with 6 straight sides. Similarly, “nona-” means $9$ in Greek, so a “nonagon” would be a $9$-sided polygon.

Here is one very confusing point. We have defined all of our polygons as two-dimensional shapes, meaning that we have included the boundary and the inside of the shape. However, we will use the word “triangle” to sometimes refer to the 2D triangle we just described, and sometimes we will use the word “triangle” to refer to the 1D boundary of the triangle. Some textbooks use language like “triangular region” to refer to the 2D shape and save “triangle” for only the 1D shape. You can use this language if you find it helpful to bring clarity to your explanations. The main point here is to be sure that you are paying attention to dimension.

Next, we have special vocabulary to talk about the side lengths of triangles.

Notice here that we said “sides” for the equilateral triangle instead of edges. We could have used the word edges, but “sides” feels a little bit like more everyday language when we start talking about triangles. Since a triangle is a polygon, we know that these sides have to be straight sides. Watch out for these ideas as we move forward! Notice that the definition for an isosceles triangle says “at least” two sides. That means that an equilateral triangle also meets the definition for an isosceles triangle! The idea that a shape can satisfy more than one definition at once is the beginning of thinking about how to compare and contrast shapes.

When we work with triangles, we often designate one of its edges as the base of the triangle. We are usually most accustomed to seeing the bottom edge labeled as the base, but any of the three edges of the triangle can be chosen as the base. Once we choose the segment called the base, we can also find a segment called the height of the triangle.

There is quite a bit of terminology to describe what we are discussing here, so in your future classroom we encourage you to look at your textbook closely. For example, some books use the term baseline to describe the extension of the base into a line. Other books will use altitude for what we just described as the height (the segment connecting the base to the opposite vertex), and will use height to refer to the length of the altitude. If these distinctions help you to write more clear explanations, we encourage you to use them!

This definition is a bit easier to understand with a picture, so let’s take a look.

The next type of definitions we have tell us about triangles whose angles have specific characteristics. Notice that when we talk about the angles in a shape, we are talking about the interior angles, or the ones made on the interior of the shape. There are also exterior angles, but you typically have to draw extra lines in order to see the exterior angles.

Our next definitions are about special kinds of quadrilaterals. Pay close attention to the details in these definitions!

Remember that quadrilaterals are still polygons, so when we talk about the “sides” here, we are still talking about edges or straight sides! Notice that we could also say that an isosceles trapezoid is a trapezoid with two pairs of equal adjacent angles.

Finally, we have one special kind of polygon that we would like to be able to recognize.

We also have shapes that have round sides instead of straight sides.

Did the answer to the dimension question surprise you? In English, we usually use the word “circle” in two different ways: first to refer to both the “outside edge” as in our definition above and second to refer to all of the 2D space inside the “outside edge”. In mathematics, we usually use the word disk to refer to the 2D space inside a circle so that we have different words for these two objects. You can choose which terminology you like, but please also use definitions in your explanations so that they are clear!

There are many other types of shapes as well, such as ellipses, ovals, and so on. If you run in to a type of shape and you aren’t sure what the definition of that shape should be, please ask us or look it up!

4 Solids

Now, let’s talk about 3D shapes.

Returning to our ideas about dimension, we usually draw lines to sketch 3D shapes on the page, but the lines themselves are only one dimensional. We can also imagine the sides of our shapes, perhaps made out of paper.

When we refer to the 3D shape, often called a solid, we are referring to the lines, the sides, and all of the space inside as well. Our friend the little bug could fly around in there!

Our 3D solids still have vertices and edges, just like our 2D shapes, but now we also have faces, which refer to the “paper sides” of the solid. Each piece of paper that you might cut out and use to build the shape would be a face.

When we think of our solid as made out of paper or another material, we may find it helpful to draw that pattern out.

We have been using the example of a cube, but we haven’t defined that yet. Let’s be a bit more specific.

Typically, we name the prism using the name of its base. So, the right prism above could also be called a right pentagonal prism, because it is a right prism with pentagons for bases. And the oblique prism could also be called an oblique rectangular prism, because it is an oblique prism with rectangles for bases.

Here is one special kind of prism.

A cylinder is like a prism, except instead of a polygon for the base, we can use any other shape. We use the same process for building: take two copies of your shape and lift one above the other without twisting so that the two copies are parallel. Connect corresponding points on the two shapes, though in this case you might have to think about connecting each point rather than connecting vertices. You may also think about wrapping a piece of fabric or flexible paper around the outside of the solid so that you can see the solid.

If we make a cylinder with a circular base, we get what you typically think about as a cylinder.

However, we can make a cylinder with many other shapes for the base, and we can make both right cylinders and oblique cylinders.

Pyramids are usually named for the 2D shape used as the base. For instance, our right pyramid above is a trapezoidal pyramid, because its base is a trapezoid. Our oblique pyramid is a triangular pyramid because its base is a triangle.

A cone is like a pyramid, except we don’t have to use a polygon for its base and can instead use any shape. In terms of an analogy, a cone is to a pyramid as a cylinder is to a prism. If we use a circle as the base and draw a right cone, we get the shape that most frequently comes to mind when we imagine a cone.

However, we could draw many other shapes for the base. Here is an example of an oblique cone with an oval base.

We also have a 3D analogue of circles and disks.

Again, we have the trouble that the sphere itself is actually two dimensional, but we also use the word “sphere” to refer to the 3D space inside the object we just defined. Unlike with circles, we don’t have another word in mathematics to solve this problem. Maybe you can come up with a good one – but as always, practice writing the definitions in your explanations! 2026-07-22 20:17:30