12.8 Contrasting and Relating the Perimeter and Area of shapes

Perimeter is different from Area: determine the area and perimeter of the shape in figure 12.89

perimeter 26 cm, and area of 21 cm sq

ex: of all shapes w/ perimeter 15 in,

a circle of circumference 15 in is the shape that has the largest area, since the C of this circle is 15 in, its radius is 15/2pi =2.4 in (approx), area = pi x 2.4^2 in sq = 18 in sq (approx)

ex: among all rectangles of perimeter 24 inches...

a square that has four sides of length 6 inches has the largest area, and this area is 6 x 6 in sq = 36 in sq

what is true in general

among all rectangles of a given fixed perimeter P, the square of perimeter P has the largest area, and every positive number that is less than the area of that square is the area of some rectangle of perimeter P

what is true in general regarding all shapes with a given fixed P

among all shapes with a given fixed perimeter P, the circle of circumference P has the largest area, and every positive number that is less than the area of that circle is the area of some shape of perimeter P.

for a given fixed perimeter, which areas can occur

answer depends on which shapes we are considering

what's a hands on way to think about perimeter

as the length of a string that wraps snugly once around the shape

by moving a 24 in loop of string to form various rectangles, you can probably tell that...

every positive number less than 36 is the area , in sq inches, of some rectangle of perimeter 24 inches

the area of the shape in figure 12.89 ( hint: units)

is the number of 1 cm by 1 cm sq it takes to cover the shape w/o gaps or overlaps

so although perimeter does not determine area...

it does constrain which areas can occur

when we calculate the perimeter of a polygon by adding the lengths of the sides around the polygon...

it is as if we had cut the string into pieces and are determining the total length of the string by adding the lengths of the pieces we are calculating the total distance around the shape by adding up the lengths of pieces that together encircle the shape

when we calculate the area of a rectangle, we...

multiply only two sides's lengths

does perimeter determine area?

no

if we consider only rectangles of a given fixed perimeter, what can we say about their areas?

of all rectangles of a given fixed perimeter the one with the largest area is a square and the greater the difference between the side lengths, the smaller the area becomes

by moving a 15 inch loop of string into various positions on a flat surface, you notice...

that every postive number less than 18 is the area, in sq in, of some shape of perimeter 15 in. ex, there is a shape that has P 1 in and area 2.7156

what if we consider all shapes that have a given fixed perimeter?

the circle is the one with the largest area

what can you notice about the rectangles in figure 12.90

the longer the perimeter, the smaller the area; the smaller the perimeter, the larger the area

the perimeter of the shape in figure 12.89 (hint: units)

the number of 1 cm segments it takes to go all the way around the shape,

calculating perimeters of polygons: why do we add the lengths of the sides of a polygon to calculate its perimeter?

the perimeter of a polygon (or other shape in a plane) is the total distance around the polygon.

for a given fixed perimeter,

there are many shapes that can have that perimeter, and these shapes can have different areas

the area of a shape is described a unit of ...

unit of area, such as sq cm

the perimeter of a shape is described by a unit of...

unit of length, such as cm,

when we calculate the perimeter of a rectangle...

we add the lengths of all four sides (or we add the lengths of two adjacent sides and multiply by 2

students sometimes get confused when deciding how to calculate perimeter and area..clarify

we add to calculate perimeter and multiply to calculate areas of rectangles