The given circle has the equation: x^2 + y^2 - 6x + 4y = 36

x^2 + y^2 - 6x + 4y = 36

=> x^2 - 6x + 9 + y^2 + 4y + 4 = 36 + 9 + 4

=> (x - 3)^2 + (y + 2)^2 = 49 = 7^2

As this is the standard form of a circle the center is (3, -2) and the radius is 7.

The area of a circle is equal to pi*r^2

=> pi*7^2

=> pi*49

**The area of the given circle is pi*49**

To determine the area of a circle the radius of the circle is required; if the radius of the circle is t, the area of the circle is A = `pi*r^2` .

For the circle x^2+y^2-6x+4y=36, first write this in the general form (x - a)^2 + (y - b)^2 = r^2 where r is the radius and (a, b) is the center.

x^2+y^2-6x+4y=36

x^2 - 6x + y^2 + 4y = 36

x^2 - 6x + 9 + y^2 + 4y + 4 = 36 + 4 + 9

(x - 3)^2 + (y + 2)^2 = 49

(x - 3)^2 + (y + 2)^2 = 7^2

The radius is 7.

The area of the circle is `pi*r^2 = 49*pi`

We'll recall the formula for the area of the circle:

A = pi*r^2

In order to evaluate the area of the given circle, we need to compute it's radius.

For this reason, we'll re-write the general equation of the circle in standard form:

(x - h)^2 + (y - k)^2 = r^2

(h,k) are the coordinates of the center of the circle and r is the radius of the circle.

We'll use the technique of completing the square:

(x^2 - 6x + 9) + (y^2 + 4y + 4) - 9 - 4 = 36

(x - 3)^2 + (y + 2)^2 = 36 + 13

(x - 3)^2 + (y + 2)^2 = 49

The coordinates of the center are (3,-2) and the radius is r = sqrt49 = 7

**The requested area of the circle is: A = 49*pi square units.**