**Given,**

• Radius of the semicircle is ‘r’.

• Area of the rectangle is maximum.

**Let us consider,**

• The base of the rectangle be ‘x’ and the height be ‘y’.

Consider the ΔCEB,

CE^{2} = EB^{2} + BC^{2}

Now the area of the rectangle is

A = x × y

Squaring on both sides

A^{2} = x^{2} y^{2}

Substituting (1) in the above Area equation

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function Z(x) has a maximum/minimum at a point c then Z’(c) = 0.

**Differentiating the equation (2) with respect to x:**

To find the critical point, we need to equate equation (3) to zero.

[as x cannot be zero]

Now to check if this critical point will determine the maximum area, we need to check with second differential which needs to be negative.

**Consider differentiating the equation (3) with x:**

Now let us find the value of

so the function Z is maximum at x = r√2

Substituting x in equation (1)

As the area of the rectangle is maximum, and x = r√2 and y = r√2/2

So area of the rectangle is

**Hence the maximum area of the rectangle inscribed inside a semicircle is r**^{2} square units.