triangle wi
th a
curv
ed base.
l
T
he length of the base is
pr
etty close t
o the length
of the arc of this sec
tor.
Approxima
tel
y
sectors are like triangles
If you pretended that a slice (sector) was a triangle, you could work
out the area of that slice and then add up all the slices.
Which technique from your Geometry Toolbox could help
you work out the pizza areas approximately?
Solution
T
r
iangle ar
ea
= 1/2 base x
height
Simil
ar
i
ty &
congruenc
e
Angles in a
quadr
ila
tera
l
add up t
o 360º.
A sec
tor is like a
Each sector (slice) is a triangle (kind of)
A sector of a circle isn’t actually a triangle with three straight edges,
but it’s pretty close. You can find the approximate area of that sector
by using the radius of the circle as the triangle height, and the arc
length as the triangle base.
Exac
tl
y equa
Height = radius
Base = arc length
226 Chapter 5
circles
Geometry Detective
8"
This pizza has a diameter of 8 inches.
Using radius to stand in for height, and arc length
to stand in for base, use the triangle area formula
to investigate how you can approximate the circle
area.…
1. By chopping the pizza base into 6 equal slices
2. By chopping the pizza base into 10 equal slices
3. By chopping the pizza base into 30 equal slices
What do you notice as you chop it into
more and more slices?
you are here 4 227
Y
ou might hav
e put something
diff
er
ent—as long as y
ou notic
ed
that the answ
er w
as a
l
number, y
ou’r
e on the right path.
We didn’t tell y
ou how man
y
decima
l pl
ac
es to use—but
if y
our answ
er
s ar
e coming
out equa
l wi
th just a couple
of decima
l pl
ac
es i
t’s a
lways
w
orth checking whether
they’re s
till equa
l wi
th mor
e
decima
l pl
ac
es, t
oo.
circle.
geometry detective solution
Geometry Detective
Solution
8"
This pizza has a diameter of 8 inches.
Using radius to stand in for height, and arc length
to stand in for base, use the triangle area formula
to investigate how you can approximate the circle
T
r
iangle ar
ea
= 1/2 base x
height
Her
e’s the triangle
area f
ormul
a f
rom y
our
Geometry T
oolbox.
Use this t
o find the triangle
(or sec
t
or) ar
ea, and then
multipl
y by the number of
slic
es w
e’r
e using t
o make the
area.…
Slices x radius x circumference x 1 = area
slices 2
w
ays the same
1. By chopping the pizza base into 6 equal slices
6 x 4 x 2
∏ x 4 x 1 = 50.2654 square inches
6 2
2. By chopping the pizza base into 10 equal slices
10 x 4 x 2
∏ x 4 x 1 = 50.2654 square inches
10 2
3. By chopping the pizza base into 30 equal slices
30 x 4 x 2
∏ x 4 x 1 = 50.2654 square inches
30 2
What do you notice as you chop it into
more and more slices??
It doesn’t seem to matter how many slices you use
to get the area approximately…the answer is always
the same?!?
228 Chapter 5
circles
No way! That’s just
freaky. How come they
all came out the same?
Jim
Joe
Frank: Well…I think I know HOW it happened, I just don’t
know WHY it happened.…
Joe: Really? Say more.…
Frank: Hmm. Well, see how the first time we multiplied by
six because we were chopping it into six slices, and then we
divided by six while calculating the arc length? They cancel
out, don’t they?
Jim: Yeah—and I guess the next one does as well—when we
chopped it into 10, and we divided and multiplied by 10.
Frank: And with the 30 pieces, too. The number of slices is
always cancelling out.
Joe: OK, I see that, but don’t you think it’s kind of weird that
we got the same answer each time, even though we were using
arc length just as an approximation for the triangle base?
Frank: Ah! OK—that’s it—that’s exactly it!
Joe: What’s it?
Frank: OK, when we have a few big slices the arc length is
really pretty different from the triangle base.
Jim: Yes, very different.
Frank: But, the more slices we chop the circle into, the nearer
the arc length is to the triangle base length. So, eventually—if
we chop the circle into like a million pieces—it’s not an
approximation anymore, it’s a perfect fit!
Jim: Yes. Cool! In fact, I think we’ve just found the formula
for the area of a circle.
Frank
Based on the pattern you found in your investigation, what do
YOU think the formula for the area of a circle might be?
T
ry s
tarting wi
th a w
ord equa
tion,
and then work i
t up in algebra if
y
ou’r
e f
eeling confident.
you are here 4 229
circle area formula
Based on the pattern you used in your investigation, what
do YOU think the formula for the area of a circle might be?
Solution
The pattern for finding area based on triangle area was:
area = slices x radius x circumference x 1
2
area = radius x circumference x 1
2
Circumference =
∏D = 2∏r
area = radius x circumference x 1
2
area = radius x 2
∏r x 1
2
area = r x
∏r = ∏r
2
You might have stopped here, which is great work, or you might have gone
a step further.…
slices
T
hese canc
el
230 Chapter 5
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.