If you know the lengths of both sides (A and B) of the rectangle, then the length of the diagonal (C) can be defined as the square root of the sum of the squares of the lengths of the sides. This follows from the Pythagorean theorem, since the diagonal in this geometric figure forms a right triangle the other two sides of which are sides of the rectangle. The diagonal of this triangle is the hypotenuse, and the sides of the rectangle are the legs. That is: C=√(A2+B2).
If the length of one side is unknown, but the known length of the other (A) and area (S) of the rectangle, the diagonal length can also be calculated. Since the area of a rectangle is found by multiplying the lengths of its sides, the unknown side can be expressed as the quotient of the square of the length of the other side. Substitute the expression obtained in the first step the formula: C=√(A2+S2/A2)=√(A⁴+S2)/A.
If you know the length of one side of the rectangle (A) and the length of its perimeter (P), the length of the second side can also be determined. Since the perimeter of a rectangle is twice the sum of the two sides, then each side can be defined as the difference between pauperisation and a length of the other side. Substitute this expression in the same formula from the first step: C=√(A2+(P/2-A)2=√(A2+P2/4-P×A+A2)=√(2×A2+P2/4-P×A).
If you know the radius of the circle (R) containing the rectangle, its diagonal will be equal to twice the radius, as the center of the rectangle and the circle in this case are the same. The line connecting the two points of the circle and passing through its centre is equal to its diameter, i.e. the two radii. And since the vertices of this rectangle lie on the circle, and the connecting diagonal passes through the centre, it is also consistent with the definition of the diameter of a circle: C=2×R.
If you know the radius of the inscribed rectangle of the circle (r), then the lengths of its sides are the same. This is a special case of rectangle called a square. To determine the length of the parties in this case as twice the length of the radius of the circle, and substituting this expression into the formula from the first step, you will get: C=√(4×r2+4×r2)=r×√8.