This post asks the reader to compare areas of an equilateral triangle, a square, and a circle when all three shapes have the same perimeter.
Question: A person has 12 inches of string. She uses the string to form an equilateral
triangle, a square and a circle. What
is the area of each shape?
A person has 24 inches of string and makes a new larger
equilateral triangle, square and circle.
Does the increase in the perimeter of the shapes impact the ratio of the
areas for the different shapes?
Answer: I discuss the area calculations for the
three shapes when perimeter is 12.
Triangle: Each side is 4. Half the base is 2. The height is (4^{2}2^{2})^{0.5}
Which is 3.4641
Note that the height can also be obtained from a trig
function 3.4641=4*sin(radians(60).
So the area of the triangle is 4*3.4641*0.5 or 6.93.
Square: The side of the square is 12/4 or 3. The area is 9.
Circle: The radius of the circle is 12/2*pi or 1.91.
The area is pi*r^{2} or 11.5.
Calculations are laid out below:
Area of Three Shapes
Given a Fixed Parameter


Triangle Area Calculations


Perimeter

12

24

Length of one side of an
Equilateral Triangle

4

8

Length of height of
Equilateral Triangle

3.4641

6.9282

Area of Equilateral
Triangle

6.9282

27.7128

Square Area Calculations


Perimeter

12

24

Length of one side of
Square

3

6

Area of Square

9

36

Circle Area Calculations


Perimeter

12

24

Pi

3.1416

3.1416

Radius of Circle

1.9099

3.8197

Area of Circle

11.4592

45.8366

Ratio of Circle to Square Area

1.2732

1.2732

Ratio of Circle to
Triangle

1.6540

1.6540

Note that changing the perimeter does not alter the ratio of
the area of the shapes.
If you enjoyed this problem you will probably also enjoy
this problem on the shape of a coke can.
Question: One Coke can is 15 centimeters long. The
second coke can is 12 centimeters long. Both coke cans hold 355 milliliters
of liquid.
What
is the radius of the top of each coke can?
What
is the volume of the surface area of the two can?
What
can is better for the environment?
Answer to Coke Can Problem is below:
Also, see
The Earth’s Actual Orbit:
http://www.dailymathproblem.com/2013/04/theearthsactualorbit.html
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