G.GMD.7 ACTIVITY #5
-
PAfiERSON
NAME:
DEVELOPING AREA FORMULAS
REGULAR POLYGONS
Regular polygons are the most specific polygon with that number of sides. Regular polygons have equal
angles, equal sides and the maximum lines of symmetry. They are only 'regular' in the sense that if we are
asked to draw a triangle we usually draw an equilateral triangle or even more so if we are asked to draw an
octagon we would immediately draw a stop sign
- the regular form of that polygon.
Regular polygons can be inscribed in circles because of their properties and because of this we need to
introduce some new terms. The center of the regular polygon is the point that is equidistant from the vertices
oftheregularpolygon. Theradiusof aregularpolygonisthedistancefromthecentertoavertex. The
apothem is the perpendicular distance from the center to a side. The height of the regular polygon is distance
between opposite sides in an even numbered polygon and the distance from a side to its opposite vertex in an
odd number polygon.
1. Draw and label a radius (r), an apothem (a) and a height (h) of the given regular polygon.
b)
a)
d)
c)
e)\
h
I
ln each regular polygon the radii divide the polygon into congruent triangles. The angle formed at the center
vertex of these triangles is found by dividing 360o by the number of sides because if there are n sides, then
there will be n central angles and they will be congruent.
2. What is the central angle for these regular polygons?
-t,'){'
-----r-- ---
3s/4
CentralAngle =
!-ZO
"
centralAngle
=
?o'
CentralAngle
= 720
CentralAngle =
600
The central angle greatly helps us with calculations within the regular polygon. You will see that these angles
are very SPECIAL (spoiler alert) and will produce some of the triangles that we have already studied.
G.GMD.7 ACNVITY #5
-
PATTERSON
3. a) ln the shaded triangle, label the radius (r), the apothem (a) and the angles within the triangle.
b) which specialtriangle is formed here?
?oo- 60o- noo
c)
A
What is the ratio apothem : radius?
l:2LzO";tzr"
d) What is the ratio apothem : half of the side of the
equilateral?
l: JA
e) What is the ratio apothem : side?
f : Z7s
f)
Use
the special right triangle and its ratios to solve the following:
Given an apothem of 5 cm, what
the length of the radius?
[O
is
Given a radius of t2 cm, what is
the length of the side?
.Zf7
c,on
Given side of L2 cm, what is the
apothem?
60
I
1:bh+abh+l_bh=
Areu
I
222
11t
-sa+-so+-sa=
222
Area
1
-a(s+s+s):Area
2
1
- (apothem)(perimeter)
2'
=
Area
Given a radius of 1 cm, what is the
length of the side?
J-3 .,^^-
crr,-
The Area of the equilateral can be thought of as the area sum
the three congruent triangles within the shape.
6Jl cm, what is
7.r"'-
cvh
Given an apothem of 10.,6 cm,
what is the side length?
zfSer.\
Given a side of
the apothem?
of
G.GMD.7 ACTIVIW #5
-
PATTERSON
4. a) In the shaded triangle, labelthe radius (r), the apothem (a) and the angles within the triangle.
b) Which specialtriangle is formed here?
u
iso -qs -
oto
o
c) What is the ratio apothem : radius?
go"
,.I '. 90"
t\
rl"
)l '.
I
t\
[: lV
,'
d) What is the ratio apothem : half of the side of the
equilateral?
[:t
ta
e) What is the ratio side : apothem?
lrt/rf)
Use
oQ"
):
t
the special right triangle and its ratios to solve the following:
Given an apothem of 5 cm, what
the length of the radius?
is
Given a radius of gJ7 cm, what is
the length of the side?
lL c^
517.*
Given side of
radius?
G
t2
cm, what is the
Given a radius of 4 cm, what is the
side length?
fz c'n
t
Q. cr.r.
The area ofthe square can be thought of as the area sum ofthe
four congruent triangles within the shape.
Lun*Lur*!m*!th: Area
2222
1111
Area
-SA+-sa+-sa+-sa=
2222
I
=Area
-a(s+s+s+s)
2
I
(apothem)( p er imeter) = Ar e a
=2'
While this is a valid formula for the square we rarely use it
because once we know a side or an apothem it is very easy to
use the basic area formula for a square, Area = (side)2.
Given a side of 8 cm, what is the
apothem?
t
clr^
Given an apothem of 2 cm, what is
the length of the side?
t
ar'.
G.GMD.7 ACTIVITY #5
-
PATTERSON
5. a) !n the shaded triangle, label the radius (r), the apothem (a) and the angles within the triangle.
b) which specialtriangle is formed here?
jo"- ro"- 10 o A
\/\/
\/\/
\/\/
\/\/
\/\/
\/\/
\/
c)
What is the ratio radius : side?
[:
"rQD,"60
@"i
t
d) What is the ratio apothem : half of the side of the
equilateral?
oc- J1 .'I
l: Jtll
e) What is the ratio side : apothem?
[:G L
f)
Use
o(L
Z; [3
the special right triangle and its ratios to solve the following:
Given a radius of 18 cm, what is
the length of the side?
Given an apothem of 8.,5 cm,
what is the length of the side?
[6
"*
I
I
ctn"
Given a radius of 4 cm, what is the
apothem length?
Given side of 1OrE cm, what is the
apothem?
Zll
ISc,,^
cv!\
The area of the hexagon can be thought of as the area sum of
the six congruent triangles within the shape.
L u,
*L
on
*!
nt,
*!un * !
u,
*! bh : Area
*L ro*1rr*!, o *L ro *! sa= Area
222222
L ro
1
:a(s+s
+.t+s
2
+s +s)
1
;z (apot he m)( per ime
t
: Area
er)
= Ar e a
Given a side of LZ cm, what is the
apothem?
6I3
,r
Given an apothem of 9 cm, what
the length of the side?
6Ecd-
is
G.GMD.7 ACTIVITY
#5- PATTERSON
5
6. a) ln the shaded triangle, label the radius (r), the apothem (a) and the angles within the triangle.
b) ls a speciattriangle is formed
c) Use trigonometry
the
slrtj6 = i
x=
Given a side length of 5 cm, what is the length of the
apothem?
f..r.36o=
to
A
A-
5.8&
Zx= stog
,<
Side=
cm (2 dec.)
t.t3
cm (2 dec.)
The area of the pentagon can be thought of as the area sum of the
five congruent triangles within the shape.
*L
ut,
22222
*L
ut,
*L th * Lbh = A*o
Lro*Lrr*!"o*1, o*!ro
22222
= Area
1
-a(s+s+s+s+s) =Arel
1
,(ano
t he
m)( p e r i me
t
er) : Ar e q
+x
X3 t.l3crn-
3
ll .7e
L nn
NO)
to solve for the following values.
Given a radius of 10 cm, what is the length of
side?
Side =
here?
7
G.GMD.7 ACTIVITY #5
- PATTERSON
6
7. Find the apothem of each regular polygon.
EQUILATERAL
a) Hexagon with radius 8 cm
8,i"i,qfs
{ [E .,*1ry
\,'rr
-t-
,,-'60"i
I
a
b)Square with side 10 cm
5
F,"
orn
c) Equilateraltriangle
ZE en
A
with radius 416cm
ret
il'
8. Find the radius of each regular polygon.
a) Square
b
with area of 64 cm2
tlZ.n trt
A
b) Equilateral triangle with apothem of 8 cm
lL
c) Hexagon
L\
"^
with apothem of lTJl
cm
I
","t
9. Find the perimeter of each regular polygon.
a) Square
{E"^
{)
b) Equilateraltriangle with radius of
j6
a7
with diagon al of lZJi cm
rz(
4.6
tru
cm
2E(E)
"*
SQUARE
zn--e
rz(3)
6+6
A
-A
--+
6 +q
HEXAGON
G.GMD.I ACTIVIW
#5-
7
PATTERSON
10. Find the AREA of each regular polygon.
a) Square
EQUILATERAL
with apothem of 4 cm
(r\(8\
,,..\
,,
/il I oIr\z
to
,t
4
,' +t
,
8
'
--t'r--'60"i
- '--
; d
'-
I
I
b) Equilateral triangle with a radius of 8 cm
LaD
zl
|
'{8J-r.^lt,r
(-)gSX,)
SQUARE
',.8-"1,r1
AP
c) Hexagon with a side of 10 cm
I^F
l, lsra)(r.)(c)
tb
I50J-3*ttrr
d) Square a radius of 14 cm
4
'i>
J! I? =.[E,fa
{iIz
31 Z,Io
,' sli
fr\-J
(
L
t,tE
rqE)(rurz)
l16Iq = 312
$,,,i
u'
/l
e) Equilateral triangle with a perimeter of i-8 cm
of
II .r.t rrr
HEXAGON
i
/,\
/
....
.'
.,,,
\
!o-o
Zl
----------t-ls\
ItnXaXs)
w;e-o
L
\ffi"""..../
,,
i.il'.
6P
G
7
\3
tr-: Y Ii
E= T-et3
t