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At a Glance - Geometry

If you don't remember them, it's a good idea to review your geometry formulas. They'll come back to haunt you all the time, and it's better if they're more like Casper the Friendly Ghost than Jasper the Grumpy Ghost Who Will Scare the Bejeezus out of You in the Middle of the Night. So strange that that Saturday morning cartoon never got off the ground.

Sample Problem

If a triangle has a height of 4 cm and an area of 20 cm2, how long is the base of the triangle?

This is a fairly straightforward question, as long as you've memorized your geometry formulas. Ahem. The area A of a triangle is given by  , so just plug in the numbers given in the problem: h = 4 and A = 20.

We're multiplying the fraction, the variable, and the 4 together. The variable we can't do anything about for the time being, but we can multiply the fraction and the 4 together to get 2.

20 = 2b

We can then divide both sides by 2 to find that b = 10 cm.

Sample Problem

In the picture below, find a formula for a2 in terms of the shaded area. Let A denote the shaded area.

The shaded area A is equal to the area of the square minus the area of the circle. There's no special formula for that part—you can figure out that part by eyeballing it. Or by smelling it, or whatever your most reliable sense is. The side length of the square is a, so the area of the square is a2.

The radius of the circle is  (that's half the square's side length), which means the area of the circle is:

Putting together all the puzzle pieces, we subtract the circle from the square.

We're almost done. The only thing left is to actually do what the question asks, which is to provide a formula for a2 in terms of A. With whipped cream and a cherry on top, if possible. We need to rearrange our current formula a bit. First, simplify the right-hand side.

There are two ways we can go from here. Technically, there are infinitely many ways we can go from here, but only two correct ones. Since time is limited, we'll only go over the correct ones.

Way 1: Factor out the a2 from each term in the right-hand side.

Divide each side by the quantity in parentheses, and here we are:

Way 2: Instead of factoring out the a2 right away, multiply both sides of the simplified formula for A by 4 to get rid of fractions.

4A = 4a2 – πa2

Which is much prettier. In fact, we'd kiss it on the mouth if it had one.

Now factor out a2 to get:

4=  a2(4 – π)

And divide by the stuff in parentheses to finish up.

This answer might look a little different from the one we got using Way 1, but they're totally identical.

Exercise 1

What is the height of a rectangular box with a volume of 144 in3, a length of 12 in, and a width of 4 in?

Exercise 2

The picture below shows one right triangle inside another. Apparently, it was cold and sought shelter. The big triangle has two sides of length 2x; the little triangle has two sides of length x. The area covered by the big triangle but not covered by the little triangle is 24. What is x?

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