what is maximum area of a quadrilateral that has a perimeter of 64 m

Question

what is the maximum area of a quadrilateral that has a perimeter of 64 m

 

Answer 1

A=lw

P=2l+2w

64=2l+2w

32=l+w

32-l=w

A=lw

A=l(32-l)

A=32l-l²

use -b/2a

-32/-2=16

you have a square that is 16 m by 16 m

A=16*16=256 square meters

FYI the maximum area is always a square.

Therefore all you had to do was 64/4=16, but I don't know what level of math you are in.

Answer 2

The quadrilateral with the maximum area for a given perimeter will be a square, so in this case divide 64 by 4 and you'll see that each length is 16, giving you an Area of 16 x 16, which is... 256 square meters.

 

If you start changing the sides to, say, 2*17 + 2*15, then your area starts to get smaller (255 in this case.)

 

If you want to test it on irregular quads, draw a random quad and measure its lengths. then draw a diagonal between 2 opposite corners to form 2 triangles.

 

Use the formula A = 1/2(b)(h) to find the area of each triangle, then add them together to get the area of the quad. You'll see that every permutation you try will result in an area smaller than 256 square meters.

 

Hope this helps!

 

Matt in New York

Answer 3

Perimeter = 3*side = 64 => side = 64/3

 

Area = ½*side a*side b *sin(angle c)

 

 
Area = ½*64/3*64/3 *sin(60⁰)