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Four equilateral triangles are constructed on the sides of a square with side length 1 as shown below. The four outer vertices are then joined to form a large square. Find the area of the large square. May 3, 2020

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If the side of an equilateral triangle is "a", the altitude of that triangle is:  a·sqrt(3)/2.

Since each of the equilateral triangles have a base of 1, the height of each of these triangles is: sqrt(3)/2.

So the distance from one vertex of the large square to its opposite vertex is  sqrt(3)/2 + 1 + sqrt(3)/2  or  1 + sqrt(3).

This is the length of each diagonal of a square.

If "d" is the length of a diagonal of a square, its area is:  A  =  ½·d2.

So, the area of this square is:  ½·( 1 + sqrt(3) )2  or  2 + sqrt(3).

May 3, 2020

#1
+1

If the side of an equilateral triangle is "a", the altitude of that triangle is:  a·sqrt(3)/2.

Since each of the equilateral triangles have a base of 1, the height of each of these triangles is: sqrt(3)/2.

So the distance from one vertex of the large square to its opposite vertex is  sqrt(3)/2 + 1 + sqrt(3)/2  or  1 + sqrt(3).

This is the length of each diagonal of a square.

If "d" is the length of a diagonal of a square, its area is:  A  =  ½·d2.

So, the area of this square is:  ½·( 1 + sqrt(3) )2  or  2 + sqrt(3).

geno3141 May 3, 2020