# 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?    Suggestions:

rectangles
squares
triangles
trapezoids
parallelograms
regular hexagons
regular octagons
circle
semicircle
quarter circle
sector of a circle

What are the dimensions of each proposed shape and what is its area?

Make area charts for those figures where different dimensions of the 100 yard perimeter are possible.

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?

Organize a table?
Make a graph?
Which rectangular region has the most area?

• from building scales models?
• from a table?
• 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?

Hint:    Construct a table areas of  isosceles triangles for base lengths from 2 to 48.  Make a graph.

Use the AM-GM Inequality to prove the maximum area of a triangle with a fixed perimeter is the equilateral triangle.

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?

What is the maximum area of a sector of a circle with P = 100?

Hint

## Areas Fence Along a Natural Boundary

What about regions built along a natural boundary? The length along the natural boundary is not part of the amount of fence that needs to be used.  For example, prove that 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?