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Head First 2D Geometry by Dawn Griffiths, Stray

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Around the
outside, through
ev
ery vertic
e
Around the inside,
t
ouching ev
ery
side just once
Radii is just the
plura
l of radius.
Meets the side
is the length of the
other t
wo sides.
T
he apothem is
the height of the
triangle…
circumradius and apothem
The circles give us the properties we need
The base of each triangle in the polygon is the same as one side of
the polygon, and the height and other two sides of the triangle are
radii of the circles inscribing and circumscribing the polygon. The
radius of the circumcircle (goes around the outside) of a polygon is
the polygon’s circumradius, and the radius of the incircle (goes
around the inside) is called the polygon’s apothem.
a
t a right angle
Circumcircle
R
adius of the
R
adius of the
Circumradius
incircle
circumcircle
Apothem
Incircle
One triangle
Base
sector
…and the circumradius
Circumradius
The whole
polygon
Apothem
Now we have everything we need
to find the formula of the area
of a polygon, so we can pick the
Base
perfect hot tub. No time to lose!
306 Chapter 7
(Hint: i
t look
s a lot like one of the intermedia
te
stages of the circle ar
ea f
ormul
a!)
Combine these.
T
hink caref
ull
y about
which area f
ormul
a
regular polygons
Polygon Formula Magnets
Rearrange the magnets to find the general formulas for the perimeter
and area of our polygon hot tubs. Use each magnet exactly once.
×
Area of one triangle ‘sector=
× ×
Total area =
×
Perimeter =
×
Total area =
y
ou’r
e using.
number of sides
number of sides
apothem
apothem
apothem
side length
side length
perimeter
side length
1/2
1/2
1/2
you are here 4 307
(Hint: i
t look
s a lot like one of the intermedia
te
stages of the circle ar
ea f
ormul
a!)
Combine these.
Base
Height
f
ormul
a w
e had f
or circle area:
1/2 x circumfer
ence x radius?
r
eminder if y
ou’re f
u
zzy.
polygon formula magnets solution
Polygon Formula Magnets Solution
Rearrange the magnets to find the general formulas for the perimeter
and area of our polygon hot tubs. Use each magnet exactly once.
Area of one triangle ‘sector=
Using 1/2 base x height
×
1/2
apothem
side length
×
×
number of sides
number of sides
apothem
apothem
side length
perimeter
1/2
1/2
side length
Y
ou can substi
tute perimeter
f
or side length x sides.
Total area =
Perimeter =
Total area =
×
×
Doesn’t this look like the intermedia
te
Don’t worry if you put the magnets in each
equation in a different order—because it’s all
multiplication the order isn’t important.
See page 230 f
or a
The apothem joins the center to the
middle of a side, bisecting it.
The apothem meets the side at a
right angle.
The circumradius joins the center to
a vertex.
The circumradius bisects the
internal angle.
308 Chapter 7

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