Find the area of a square if its diagonal is 1.

The diagonal divides the square into two identical isosceles (their two legs are the sides of the square, and all sides of the square are equal) right-angled triangles.
Let’s denote the equal legs of one of the resulting triangles as x and find the value of x by the Pythagorean theorem:
1 ^ 2 = x ^ 2 + x ^ 2;
2x ^ 2 = 1 ^ 2;
x ^ 2 = 1/2;
x = √ (1/2) = 1 / √2 = √2 / 2 (conventional units).
The area of a square across its side is found by the formula:
S = a ^ 2,
where S is the area of the square, and is the side of the square.
S = (√2 / 2) ^ 2 = 2/4 = 1/2 (conventional square units).
Answer: S = 1/2 conventional square units.