Textbook

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, pyramids 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.

**Example 1**

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

(spoiler)

All pyramids (and cones) have volume $3Bh $, 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 $391 $ 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 $34_{2} :31_{2} $, a.k.a. $4_{2/3}:1$ or $316 :1$.

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