1) Find the side of the square.
In the problem statement, we are given the radius of a circle circumscribed around a square.
We can find the formula for the radius of a circle circumscribed around a square from the Pythagorean theorem, since the diagonal of a square is the diameter of the circumscribed circle, we denote it by the letter d.
The side of the square is denoted by the letter a.
A square is a quadrangle in which all sides are equal.
Based on the Pythagorean theorem:
d² = a² + a² = 2a²,
d = √ (2а²) = а√2
The radius of the circle is half the diameter:
R = d / 2,
where R is the radius of the circumscribed circle.
R = (a√2) / 2 = a / √2.
The formula for the radius of a circle around a square:
R = a / √2.
a = R√2,
a = 32√2
2) Find the area of the square. Let’s write the formula for the area of a square:
S = a²,
where a is the side of the square, S is the area of the square.
Substituting the value of the side of the square into the formula, we find the area:
S = (32√2) ² = (1024 × 2) = 2048 sq. Units.
Answer: the area of a square is 2048 square units.
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