regular polygons
Two tubs down, five to go
If there were only two tub designs your work would be pretty much done,
but there are another five shapes of hot tub to compare in order to find the
perfect hot tub for that backstage area at the rock festival.
Let’s take
the tr
iangle
hot tub nex
t.
Our complete summer range..
.
A
l
l
St
r
H
o
t T
ubs
T
r
i
-
c
u
z
i
C
h
i
l
l
-
ou
t
-
c
or
n
er
H
ex
-
i
t
-
u
p
7
t
h
H
eav
en R
el
ax
i
v
8
9
s
i
d
es
M
ag
i
c
C
i
r
c
l
e
T
ubs can be an
y size,
all depths 0.5m
19.6
17.3
Congruenc
e
Pythagor
ean
T
heor
em
N, Z, and
F angles
Area =
1/2 base x
height
Hero’s
f
ormul
a
Let’s work on the triangle tub next. You need to work
backward from area to find side length. Which tool from
your Geometry Toolbox is the best one for this job?
you are here 4 289
hero’s formula to the rescue
Both of these dea
l
wi
th tr
iangle ar
ea.
Let’s work on the triangle tub next. You need to work
backward from area to find side length. Which tool from
your Geometry Toolbox is the best one for this job?
Solution
Hero’s formula is A = s(s-a)(s-b)(s-c).
It finds area in terms of the sides of the
triangle, so it’s a good fit with our problem:
finding perimeter from area.
Y
ou
can
use
this f
ormul
a but
y
ou hav
e t
o do qui
te a lot
of gnarl
y a
lgebra t
o get t
o
the sides. Not our f
avor
i
te
w
ay t
o sp
end time....
Ringing bells but kind of
sketch
y? C
heck out page 196
t
o r
emind y
our
self how t
o
use Hero’s formula.
Congruenc
e
Pythagor
ean
T
heor
em
N, Z, and
F angles
Area =
1/2 base x
height
Hero’s
f
ormul
a
All your toolbox tools are arranged by
subject in a downloadable PDF at
headfirstlabs.com!
Working smarter, not harder is often about choosing the
right tool for the job. Go to www.headfirstlabs.com/geometry to
grab yourself a handy printable reference sheet of
your geometry tools grouped by topic.
290 Chapter 7
Hero’s f
ormul
a a
lr
eady has a letter s in
i
t, so let’s ca
ll our tr
iangle side “t.”
Hot Tub
regular polygons
Pool Puzzle
Use the algebra parts in the pool to find
a special version of Hero’s formula
for your equilateral triangle-shaped
hot tub. (You should use each part
exactly once.)
t
t
t
s = a + b + c
Hero’s formula: A = s (s-a) (s-b) (s-c) &
2
a, b, and c are sides of a
triangle. In our case all the
A = ......................................... &
s = .......
sides are t.
A = .........................................
A = .....................
A = .....................
2
(s-t) = ............... = ..............
subs
ti
tute f
or s
subs
ti
tute f
or (s-t)
Squar
e roots hints
This is the formula
for the area of an
equilateral triangle.
t
4
= t
2
16 = 4
3 = 3
Use these f
or
the final step.
Note: each thing from
the pool can only be
used once!
3t
2
4
3t
4
16
3t t t t
2 2 2 2
t
2
3t
3t - t
2
s s-t s-t s-t
you are here 4 291
Hero’s f
ormul
a a
lr
eady has a letter s in
i
t, so let’s ca
ll our tr
iangle side “t.”
the final step.
Hot Tub
hot tub puzzle solution
Pool Puzzle Solution
Use the algebra parts in the pool to find
a special version of Hero’s formula
for your equilateral triangle-shaped
hot tub. (You should use each part
exactly once.)
t
t
t
s = a + b + c
Hero’s formula:
A = s (s-a) (s-b) (s-c) &
2
a, b, and c are sides of a
&
s = .......
.........................................
s s-t s-t s-t
3t
triangle. In our case all the
A =
sides are t.
subs
ti
tute f
or s
2
3t - t
t
(s-t) = ............... = ..............
2
2
.........................................
3t t t t
A =
2 2 22
3t
4
A = .....................
This is the formula for the
16
area of EVERY equilateral
triangle.
3t
2
A = .....................
4
subs
ti
tute f
or (s-t)
Squar
e roots hints
t
4
= t
2
16 = 4
3 = 3
Use these f
or
292 Chapter 7
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