# Find the area of a square circumscribed around a circle of radius 32

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.