In polar coordinates, r represents the distance from the origin (0,0) to the point (x,y). So, r=4 means the distance between the origin and the point is 4 units.
Now, in rectangular coordinates, the same distance is represented by the Pythagorean theorem which is a^2 + b^2 = c^2, where c is the hypotenuse of a right-angled triangle, and a and b are the other two sides. The hypotenuse in this case would be 'r', and a, and b would be 'x' and 'y' respectively.
Therefore, we can rewrite the Pythagorean theorem in terms of x, y, and r:
r^2 = x^2 + y^2
Given that r = 4, we substitute 4 in place of r. Hence, we get:
(4)^2 = x^2 + y^2
Solving this gives us the rectangular equation for the polar equation r = 4, which is:
x^2 + y^2 = 16