LESSON 107
Maximizing Area (Fixed Perimeter)
Perimeter and Area Review
Perimeter
Rectangle 𝑃 = 2(𝑙 + 𝑤)
Regular 𝑃 = 𝑛 𝑠
Circle 𝐶 = 𝜋𝑑 or 2𝜋𝑟
There are more…
This is just at quick
review…
Area
Triangle 𝐴 = 12𝑏ℎ
Parallelogram 𝐴 = 𝑏ℎ
Square 𝐴 = 𝑠 2
1
Regular 𝐴 = 2𝑎𝑃
Circle 𝐴 = 𝜋𝑟 2
Trapezoid 𝐴 = 12 𝑏1 + 𝑏2 ℎ
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
2 feet
b)
6 feet
c)
9 feet
d)
14 feet
e)
17 feet
What are the formulas
for the perimeter and
area of a rectangle?
𝑃 = 2 𝑙 + 𝑤 ; 𝐴 = 𝑙𝑤
a)
Let’s do some work with
these formulas to make
the rest of the problem
easier.
36 = 2(𝑙 + 𝑤)
18 = 𝑙 + 𝑤
18 − 𝑤 = 𝑙
𝐴 = 𝑙𝑤
𝐴 = (18 − 𝑤)𝑤
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
a)
b)
c)
d)
e)
2 feet
6 feet
9 feet
14 feet
17 feet
a)
𝐴 = (18 − 𝑤)𝑤
𝐴 = 18 − 2 2
𝐴 = 16 2
𝐴 = 32 𝑓𝑡 2
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
a)
b)
c)
d)
e)
2 feet 𝐴 = 32 𝑓𝑡 2
6 feet
9 feet
14 feet
17 feet
b)
𝐴 = (18 − 𝑤)𝑤
𝐴 = 18 − 6 6
𝐴 = 12 6
𝐴 = 72 𝑓𝑡 2
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
a)
b)
c)
d)
e)
2 feet 𝐴 = 32 𝑓𝑡 2
6 feet 𝐴 = 72 𝑓𝑡 2
9 feet
14 feet
17 feet
c)
𝐴 = (18 − 𝑤)𝑤
𝐴 = 18 − 9 9
𝐴= 9 9
𝐴 = 81 𝑓𝑡 2
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
a)
b)
c)
d)
e)
2 feet 𝐴 = 32 𝑓𝑡 2
6 feet 𝐴 = 72 𝑓𝑡 2
9 feet 𝐴 = 81 𝑓𝑡 2
14 feet
17 feet
d)
𝐴 = (18 − 𝑤)𝑤
𝐴 = 18 − 14 14
𝐴 = 4 14
𝐴 = 56 𝑓𝑡 2
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
a)
b)
c)
d)
e)
2 feet 𝐴 = 32 𝑓𝑡 2
6 feet 𝐴 = 72 𝑓𝑡 2
9 feet 𝐴 = 81 𝑓𝑡 2
14 feet 𝐴 = 56 𝑓𝑡 2
17 feet
e)
𝐴 = (18 − 𝑤)𝑤
𝐴 = 18 − 17 17
𝐴 = 1 17
𝐴 = 17 𝑓𝑡 2
Determine the area of a rectangle with
a perimeter of 36 feet if the width is:
2 feet 𝐴 = 32 𝑓𝑡 2
b)
6 feet 𝐴 = 72 𝑓𝑡 2
c)
9 feet 𝐴 = 81 𝑓𝑡 2
d)
14 feet 𝐴 = 56 𝑓𝑡 2
e)
17 feet 𝐴 = 17 𝑓𝑡 2
Which rectangle had the
maximum area?
The Square
a)
What kind of a polygon
is a square?
Regular
So if you want to
maximize area with a
fixed perimeter, what
type of polygon should
you choose?
Regular
The following regular polygons have a
perimeter of 60 inches
Find their areas
1
2 20 17.32 = 173.2
152 = 225
1
8.26 60 = 247.8
2
1
2 8.66 60 = 259.8
What did you notice about
the area as the number of
sides increase?
Area increases
Questions?
What shape
has the
maximum
area?
A circle
Which pizza would you choose if the length
of crust is about 32 inches? Why?