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regular polygons
What do we need to know about
the polygon triangles?
We know that there are the same number of triangles as sides of
the polygon, and we know that all the triangles are congruent so
we just need to find the area of one. But what else do we need to
know to find the triangle’s area, so that we can combine them to
find the total polygon area?
b
b
h
a
c
Area =
1/2 base x
height
Hero’s
f
ormul
a
area?
base?
height?
side a?
side b?
side c?
W
her
e do w
e get
these?
W
her
e do w
e get
these?
Well, we used the fact that a circle is
like a polygon to find the circle area formula.
So, doesn’t that mean that a polygon is like a
circle, too? Or at least there’s a connection
between circles and polygons and these
triangle segments? Could that help?
That’s an interesting connection.
If a circle is like a polygon, the reverse must also be true
on some level: a polygon is (kinda) like a circle. Maybe
this idea can help us find what we need to calculate the
polygon area and choose the perfect hot tub?
you are here 4 303
geometry detective
Geometry Detective
Your mission is to investigate how you could find the missing information
needed to use either of the triangle area formulas by using circles fitted
to a polygon.
W
e used these circles
to find the tr
iangle
c
enters in C
hapter 4.
The polygons below are drawn with either their circumscribing circle
(around the outside, touching every vertice) or their inscribing circle
(within the polygon, touching every side).
Draw or sketch on the polygons—we’ve given you some spares so don’t
be shy!
T
hese circles inscribe
the p
olygon.
T
hese circles
circumscribe the
p
olygon.
Hint: Start by dividing the polygons into equal triangles.
You’re looking for base, height, and side length of those.
304 Chapter 7
regular polygons
Need a 10-minute time-out?
If you’re feeling kind of fuzzy about
how the polygon triangle properties relate to
the circles, try having a 10-minute brain-break,
then look again before you flip the page.
you are here 4 305

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