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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