## Wednesday, March 8, 2017

### Area of three shapes with same perimeter

Area of three shapes with 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 (42-22)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*r2 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/the-earths-actual-orbit.html