If we would live in a 2-dimensional world, we would get pretty bored.

Fortunately, all physical objects that we’re seeing and using on an everyday basis —computers, cars, phones, shoes— are existing in three (3) dimensions.

**Next lesson: **The Pythagorean Theorem

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The following transcript is provided for your convenience.

They all come with length, width, and height and even thin objects such as pieces of paper are 3-dimensional. Now, a piece of paper’s thickness may be just the fraction of an inch or of a millimeter, but there is thickness.

In geometry, it is very common to see 3-dimensional figures or shapes and in mathematics, we call the flat side of a 3-dimensional figure a face. We call shapes “Polyhedrons” if they come with four (4) or more faces, and each face is a polygon. These shapes include cubes, pyramids, and prisms. Sometimes, we may even notice a single figure which is the composites of two (2) of these figures. Let have look at a few common polyhedrons.

**Identifying Solids**

Our first set of solids includes rectangular bases. Take a look at this table below that is showing each figure in both transparent and solid forms.

Do you notice that we use different names for these figures? Cubes are different than squares although sometimes, they are confused with one another. Cubes have three (3) dimensions, whereas squares only have two. Likewise, we would describe shoeboxes as rectangular prisms (not simply rectangles), and Egypt’s ancient pyramids as…yes indeed, as pyramids and not as triangles!

In the following set of solids, all figures have circular bases.

Let’s compare pyramids and cones. Do you notice that pyramids have rectangular bases and triangular, flat faces? Cones have circular bases and rounded, smooth, bodies.

And finally, let’s take a look at a pretty unique shape, the sphere.

You can find numerous spherical objects everywhere around you such as soccer balls, baseballs, or tennis balls. These all are common items. They may not all be perfectly spherical objects, but they, generally, are referred to as a sphere.

**Volume**

Do you remember that perimeters measure one (1) dimension (length), and that areas measure two (2) dimensions (length & width)? Now, to measure how much space a 3-dimensional shape or figure is taking up, we are using another measurement that’s called “volume”.

If we want to visualize what exactly “volume” is measuring, look back at transparent images of rectangular prisms mentioned earlier or think of the empty shoebox. Now imagine that we stack identical cubes inside this box in a way that we’ll have no gaps between these cubes. Just imagine that we would fill up the whole box in this way. If we then would count the total number of cubes inside the rectangular prism, we would have the volume.

The measure for Volume is cubic units. The shoebox that’s illustrated above can be measured using cubic inches (generally represented as inches^{3} or in^{3}), whereas the appropriate measure for Egypt’s Great Pyramid would be in cubic meters (meters^{3} or m^{3}).

If we want to find a geometric solid’s volume, we may create a transparent version of that solid, make a bunch of one by one (or 1x1x1) cubes, and stack these carefully inside the solid. However, this would be taking a long period of time! A far easier way to come up with the volume is to get familiar with a few geometric formulas and use these, instead.

Okay, so let’s go through our geometric solids one more time and list the right volume formula for each of them.

When looking through our list below, we may see that some of these volume formulas are looking similar to the corresponding area formulas. If we want to find a rectangular prism’s volume, we need to find the area of the base first, and then multiply it by the height.

Keep in mind that all cubes are actually rectangular prisms, so our formula for finding a cube’s volume is the area of the cube’s base times its height.

Now, let’s take a look at solids that come with circular bases.

Well, here we see that number π again.

A cylinder’s volume is the area of the base (π r^{2}) times the height (h).

Let’s compare the formula for the volume of cones ( ) with the formula for a pyramid’s volume: ( ). Well, we can see that the numerator of cone formulas is the same as the volume formula for cylinders, and that the numerator of pyramid formulas is the same as the volume formula for rectangular prisms. Then we divide each by three (3) to come up with the volume of our cone and our pyramid. Looking for similarities and patterns in those formulas may help us remember which formula is referring to what given solid.

And finally, the formula we use for spheres is shown below. You should notice that here, the radius is cubed and not squared. Also, we multiply the quantity π r^{3} by 4/3.