3-D geometry (cloned) | Geometry (cloned)

3-D geometry

Here are a few things that I just want all AMC competitors to know:

The formulas for common three-dimensional shapes, including prisms with arbitrary bases, cones with arbitrary bases, and spheres
The fact that all two- and three-dimensional measurements of arbitrary similar shapes use the square and cube of their relative scale factors, respectively.

Examples

Example 1

What’s the volume of a cone whose base is a regular seven-pointed with area $7$ , and whose height is $13$ ?

(spoiler)

All cones have volume $\frac{B h}{3}$ , irrespective of the shape of the base.

If you got hung up on the star, you need the drillwork in this unit.

But if you got $\frac{91}{3}$ without a hiccup, you’re good.

Example 2

I have two spherical balloons, one of which holds four times as much air as the other. What’s the ratio of their surface areas?

(spoiler)

If you used the volume and/or surface area formulas for spheres, then I’m afraid you fail this test.

And if you got an integral ratio, then you were probably careless – also a fail, I’m afraid.

The right way – and I’m afraid there is only one right way in this case – is to recognize that a volume ratio of $4 : 1$ means a radius ratio of $34 : 31$ , and a surface area ratio of $3 4^{2} : 3 1^{2}$ , a.k.a. $4^{2/3} : 1$ or $316 : 1$ .

What to do

If everything above was old news to you, then this section should be no problem. Click “complete” below, and we’ll add all the relevant quizzes to your deck. Then, during your short daily practice, we’ll occasionally quiz you on this knowledge in a way that etch it into your memory for good.
If some or all of this was new to you, then you should still click “complete” below, and add it to your quizzes, but I recommend against adding anything else for today. Instead, work with these new cards for today, and consider adding more fresh knowledge tomorrow. You might also do a bit of research to refresh (or learn for the first time) the rules given above.