+0  
 
0
80
1
avatar

What is the ratio of the area of a square to the area of an equilateral triangle if they have the same perimeter?

 May 11, 2020
 #1
avatar+21953 
0

Let each side ot the equilateral triangle be x.

Then, its perimeter is 3x.

The formula for the area of an equilateral triangle is:  Area  =  ( sqrt(3)/4 )·side2.

For this triangle:  Area  =  ( sqrt(3)/4 )·x2.

 

Since the perimeter of the equilateral triangle is 3x, the perimeter of the square must also be 3x.

Perimeter of the square = 4·side.

For this square:  3x = 4·side, so each side = (3/4)·x.

The formula for the area of a square is:  Area  =  side2.

For this square:  Area  =  [ (3/4)·x ]2  =  (9/16)·x2.

 

Ratio of the area of the equilateral triangle to the area of the square

     =     ( sqrt(3)/4 )·x2   /   (9/16)·x     =     ( sqrt(3)/4 )   /   (9/16)  =  4·sqrt(3) / 9

 May 11, 2020

73 Online Users

avatar
avatar
avatar
avatar