# geometric shapesAssume that you have exactly 100 feet of chain link fence material that you will use to make a pen for your dog. Also, assume that you can use any of the plane geometric shapes...

geometric shapesAssume that you have exactly 100 feet of chain link fence material that you will use to make a pen for your dog. Also, assume that you can use any of the plane geometric shapes Which shape should you choose to give your dog the largest total area? Illustrate that your choice provides a greater area than at least 2 other possibilities.

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This is fascinating! Who knew that providing your dog with a circular kennel would get him the most space? I will play devil's advocate though, and suggest that dogs like corners. They want to have a place to put their back into and feel safe. So the hexagon seems the best bet.

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I will examine a 100-sided regular polygon.  If you split it into 100 pizza slice triangles, then each one has a base of 1 foot (with central angle of 3.6 degrees), and the other two sides are equal.  Split it down the middle to form two right triangles, each with base 0.5 across from an 1.8 degree angle.

To find the area, we need the height of the triangle (and since this is a right triangle, we need the other leg).  Since the tangent ratio gives us opposite over adjacent, we have

tan(1.8)=0.5/x  ---> x = 0.5/tan(1.8)

Now we have the height of each pizza slice, so we can find the area:

A = 1/2*1*x = x/2

And multiplying this by 100 gives us the polygon's area:

A = 50x = 795.51ft^2... pretty close to the circle!

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The ratio of the area of a closed figure to the perimeter of the figure increases as the number of sides is increased. This can be illustrated by starting with a figure with 3 sides, (as it is not possible to have a closed figure with 2 sides)

The area of an equilateral triangle with a perimeter of 100' is given by ((sqrt 3)/4)*(100/3)^2 = 481.12 square feet.

For a square, the area increases to 25^2 = 625 square feet

Similarly, the area of enclosed by the figure formed would increase and reach a maximum when the number of sides is infinity, which is a circle.

The area of the circle is pi*(100^2/4*pi^2) = 100^2/4*pi = 795.77 square feet

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A circle gives the largest area.

Circumference of a circle equals 2pi(r)

Area of a circle is pi(r^2).

Radius of circle with 100 ft circumf. is 100/2pi = 50/pi

Area of circle with 100 ft. circumf. = pi (50/pi)^2 = 50^2/pi = 2500/3.14 = 796 square feet.

Now take a square enclosure, sides 25 feet each.  Area is 25 x 25 = 625 square feet.

Now take an equilateral right triangle:  Area of triangle = 1/2 base x height.   For this example, sides 33.3 feet each, area is 1/2 (33.3)(33.3) = 554 square feet.

The circle gives the largest area, compared to square or triangle.

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Okay, since the first poster took circles and squares, I will look at hexagons and a pentagon.

A hexagon has 6 sides. Each side will take 16.6 feet of fencing ( 100/6=16.6). A hexagon created by using 100 feet of fencing will have an area of 715.93 square feet.

A pentagon has five sides. Each side of fencing will be 20 20 feet long (100/5=20). Therefore, the area of a pentagon with a 20 foot long side would be 688.19 square feet.

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A circle with a 100' circumference will give you an enclosed area of 796 square feet.  Circumference is equal to 2 pi radius, so

C = 2(pi)r

100 = 2(pi)r

15.92 = r

the Area of a circle is (pi) r^2, so

A = (pi) r^2

A = (3.1416) (15.92 feet)^2

A = 796 sq feet.

In contrast, consider a square, whose area is length * width.  Each side must be 25 feet, so

A = l * w = 25 feet  * 25 feet  = 625 sq feet.

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The post #8 was submitted inadvertently and may please be considered deleted. Complete answer is as under:

Very interesting question with similarly interesting answers but all have ignored 2 basic things in the answer:

1.  "you have exactly 100 feet of chain link fence material that you will use to make a pen for your dog", saying that the fence has to be made.

2. " Illustrate that your choice provides a greater area than at least 2 other possibilities", that there has to be 3 possibilities in all.

None of the earlier posts qualify both prerequisites. Only post #5 gives 3 options.

From construction point of view, equilateral triangle, square, hexagon, octagon and circle are the simpler options with circle being the simplest to make. In this case the radius = 15.92 feet and area of about 796 square feet as given in the earlier posts. This option also provides the largest area.

In comparison to this, a square with 25 feet side provides an area of 625 square feet.

The the other option can be taken as a hexagon with sides = 100/6 feet

Area of the hexagon = 6*0.5*r*cos(30)*r*sin(30) = 722 square feet

Considering the above tree options, a circle gives the largest area with minimum effort as compared to square or hexagon. However, taking in to consideration the dogs comfort as per post #7, a hexagon may be considered the best option by sacrificing 74 sq ft area. It is the second best option to construct any polynomial.

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Very interesting question with similarly interesting answers but all have ignored 2 basic things in the answer:

1.  "you have exactly 100 feet of chain link fence material that you will use to make a pen for your dog", saying that the fence has to be made.

2. " Illustrate that your choice provides a greater area than at least 2 other possibilities", that there has to be 3 posibilities in all.

None of the earlier posts qualify both prerequisites.

From construction point of view, equilateral triangle, square, hexagon, octagon and circle are the simpler options with circle being the simplest to make. In this case the radius = 15.92 feet and area of about 796 square feet as given in the earlier posts. This option also provides the largest area.

In comparision to this, a square with 25 feet side provides an area of 625 square feet.

The the other option can be taken as a hexagon.