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.
+23246 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).
+23246 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).
