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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