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