Head First 2D Geometry

regular polygons

Rock stars—high maintenance?

Uh oh. The band is getting demanding. They’ve seen the All Star

Hot Tubs brochure lying around and think there’s a couple of

designs that look pretty uncool.

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you are here 4 311

Which hot tub do you want to order?

exercise solution

The HEX-IT-UP tub with six sides.

Which hot tub do you want to order?

Q:

A circle is just a polygon with

infinite sides? Have you gone a bit

philosophical or what?

A:Some math geeks get a bit upset

about this definition of a circle, but if you

could draw enough sides then you’d be

approaching a circle. The whole point of

infinity is that you never get there, but you

get close enough for this to be the basis of

computer graphics.

Q:

Computer graphics? Polygons?

How does that work?

A:

To show 3D curvy worlds, all video

games rely on triangles, which build into

polygons, which fool our eyes into believing

we’re seeing circles and curves.

Q:Ah! Yeah, when I’ve zoomed

in real close to something in a game

sometimes I’ve seen that it’s made out

of flat shapes.

A:That’s it—exactly.

Q:

Circumradius sounds a lot like

radius, but in the comparison of the area

formulas you compared apothem to

radius. That’s kind of confusing!

A:Actually a regular polygon’s

circumradius is sometimes just referred

to as its radius. But if you think about it, a

circle’s apothem and its circumradius can

both be equal to its radius (if the apothem

is going right to the edge). So when we

compared the apothem and the circle’s

radius, we were comparing an apothem to

an apothem!

Q:

Does the polygon area formula

work for all regular polygons? Even

squares and triangles?

A:The polygon area formula works for

any regular polygon—whether it’s got 3

sides or an infinite number! But you’ve got

some much sharper tools in your tool box for

finding square and triangle areas, and they’ll

always be quicker and easier to use, so it’s

best to keep this one for regular polygons

with five sides or more.

Q:

Regular polygons only? This

area formula won’t work with irregular

polygon?

A:Sadly not. Irregular shapes are harder

to deal with anyway. When you split them

into triangles the triangles aren’t congruent.

Very high maintenance. Luckily they don’t

come up in exams much!

Q:

Apothem. Let me guess, that

means in an algebra version of the

formula that would be an “a”? But we’re

already using “a” for area—why couldn’t

they have named it something with a

different first letter?

A: Good question. We don’t have an

answer—except that if you remember your

formulas as words rather than just letters

then it doesn’t matter so much. If you’re

sitting a test and you want to demonstrate

that you know the formula, write it out in

words and then make it clear which letters

you’re picking as stand-ins. Whether you

use a, A, s, or t for something is less

important than what it means. It’s also easier

to work out whether a formula applies to a

problem when you remember your formulas

as words and not just letters.

312 Chapter 7