Exploration: Areas of Figures with a Fixed Perimeter
Getting started
Suppose one had 100 yards of fencing to enclose a garden.
What shapes could be enclosed?
rectangles
squares
trapezoids
parallelograms
quadrilaterals
hexagons
octagons
circle
semicircle
quarter circle
sector of a circle
What are the dimensions of each proposed shape and what is its
area? Make some charts.
Make scale models of the different shapes using
a loops of string all of the same length. Compare the areas visually.
Rectangles
What rectangular regions could be enclosed? Areas?
from a graph plotted from the table?
from a graph generated by a formula?
from algebra, using the arithmetic mean-geometric mean inequality?
Triangles
What triangular region with P = 100 has the most area?
Find all five triangular regions with P = 100 having integer sides
and integer
area. (such as 29, 29, 42)
Other Regular Polygons
What is the area of a regular hexagon with P = 100?
What is the area of a regular octagon with P = 100?
What is the area of a regular n-gon with P = 100?
Make a table for n = 3 to 25.
Make a graph.
What happens to 1/n(tan 180/n) as n increases?
What if part of the fencing is used to build
a partition perpendicular to a side?
Consider a rectangular region
- with one partition?
- With 2 partitions?
- With n partitions? (There is a surprise in this one!!)
- What if the partition is a diagonalof the rectangle?
What is the maximum area of a sector of a circle
with P = 100?
(Here is another surprise!!!) Hint
What about regions built along a natural
boundary? For example the maximum for both a rectangular region
and a triangular region built along a natural boundary with 100
yards of fencing is 1250 sq. yds.
But, the rectangle is not the maximum area four-sided
figure that can be built. What is the maximum area of a four-sided
figure where three sides are formed by the fencing and one side
by the natural boundary?
Other situations
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