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

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