# Find the area of the square if square and an equilateral triangle have same perimeter?

+1 vote
524 views

A square and an equilateral triangle have the same perimeter. If the area of the triangle is 16√3 , what is the area of the square?

posted Sep 17, 2014

## 6 Solutions

Let the side of Equilateral triangle be t
And of square be s
By first condition
4*s = 3*t
Area of equilateral triangle = √3/4 * (side^2) = √3/4 * (t^2)
16√3/4 = √3/4 * (t^2)
So t^2 = 16*4
t = 8
Now
4*s = 3*8
s = 6
So area of square = side^2 = 6^2 = 36

solution Feb 3, 2015

Let x as side of triangle, y as side of square.
hence 3x = 4y
area of triangle = 1/2 bh, b = x , h = √3 x /2
hence √3x^2 / 4 = 16√3
x = 8
3x = 24 = 4y
y = 6
area of square = y^2 = 36.............

solution Sep 19, 2014

let side of the triangle be b
then area of the triangle is 0.5*b*0.5*b*sqrt(3)
given area of the triangle is 16*sqrt(3)
from the above two b=8
therefore side of the square is 6 and area 36

solution Jul 5, 2015

Suppose you have perimeter as 12x then each side of triangle would be 3x and for square it would be 4x.

now (√3/4)*(4x)^2 = 16√3
or (4x)^2 = 64 = 8^2
or x = 2

area of the square is (3x)^2 or 36

solution Sep 17, 2014
if we  have perimeter as 12x then each side of triangle would be 4x and for square it would be 3x.
Thanks  for pointing corrected :) my answer.
The two perimeters must be equal, saying that 4x = 3x is wrong (except if x is zero). How do you get x=2 ?

Let the side of the equilateral be s, then area of the equilateral ▲= (√3s²)/4 = 16√3 => s² = 64 → s = 8

Perimeter of the equilateral ▲= 3s = 24

Perimeter of the square = 4a = 24, hence a= 6.

Area of the square a² = 6² = 36

solution Jun 21 by anonymous

The area of square would be 32.

solution Sep 17, 2014

Similar Puzzles
+1 vote

A triangle, a square, a pentagon, a hexagon, an octagon and a circle all have an equal perimeter, which one has the smallest area?