# Topics Examples

#### 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...

#### Surface Area of Prisms and Cylinders

We've calculated the lateral area for both prisms and cylinders. Lateral area is part of surface area, but it doesn't tell the whole story. We need the bases, too.No, not those bases.The surface ar...

#### Surface Area of Pyramids

No offense to prisms and cylinders. They're great. But we don't eat our ice cream out of prisms while touring the Egyptian Cylinders of Giza. What we're saying is that there's no substitute for pyr...

#### Surface Area of Cones

We've covered pyramids (literally, since we found the surface area), and now it's time to cover cones (literally, since we'll find the surface area).A right cone is a cone where the axis is also th...

#### Surface Area of Spheres

If you play basketball, eat oranges, or have eyeballs (which are sort of necessary to read this), you've most likely encountered spheres at some point in your life. They're such common everyday obj...

#### Volume of Prisms

For prisms, the fastest way to find the volume is to multiply the area of the base (whatever it is) times the height.V = BhThat's all volume is: base times height
Sample ProblemWhat is th...

#### Volume of Cylinders

The volume formula works not only for prisms, but for cylinders, too. The only difference is that the base is circular instead of triangular or rectangular or whatevular. If that's true (which it t...

#### Volume of Pyramids

Although we might wonder how much tinfoil it takes to cover the Egyptian pyramids, finding out how many mummies fit inside it would be much more interesting. Provided they don't form an undead army...

#### Volume of Cones

The same formula we used for pyramids (V = ⅓Bh) applies to cones, only the base isn't a polygon anymore. It's a circle. Formula-wise, that means we replace B with πr2. That's it.
Sampl...

#### Volume of Spheres

Everyone knows the moon isn't really made of cheese. But what if it were? Picture it. Sure, we'd have a bit of an issue with the ocean tides, but think of all that cheese. We could end world hunger...

#### Congruent and Similar Solids

By now, we've earned quite a bit of street cred working with surface area and volumes. You could throw us any shape and we'd give you its surface area, volume, and even its pants size.What we need...

#### The 3D Coordinate System

Two-dimensional shapes have an x-y plane to go home to at the end of the day, and what do the solids have? Nothing. Three-dimensional shapes have been homeless for such a long time, they've begun t...