In which circle can the rectangle of the largest area be inscribed if its perimeter is 56 cm.

If you draw a rectangle into a circle, the diagonals of the rectangle will be the diameters of the circle.

The area of ​​a rectangle can be calculated using the formula: S = 1/2 * d1 * d2 * sina (where d1 and d2 are the diagonals of the rectangle, and is the acute angle between them).

Since the diagonals of the rectangle are equal, the formula takes the form: S = 1/2 * d² * sina.

The sine of an acute angle can be greater than zero and less (or equal to) one.

For the area to be the largest, you need to substitute the value of the largest sine in the formula, and this is one. If sina = 1, then the angle between the diagonals is 90 ° (sin90 ° = 1).

A rectangle whose diagonals intersect at right angles is a square.

We calculate the length of the side of the square through the perimeter: 56: 4 = 14 (cm).

We calculate the length of the diagonal using the Pythagorean theorem (two adjacent sides of the square and the diagonal form a right-angled triangle):

d = √ (14² + 14²) = √ (2 * 14²) = 14√2 (cm).

This means that the diameter of the circle is 14√2 cm, therefore, the radius is R = 14√2: 2 = 7√2 cm.

Answer: in a circle with a radius of 7√2 cm.