What is the value of "r" in the circle formula (x-h)²+(y-k)²=r²?

In the circle formula ((x-h)^2 + (y-k)^2 = r^2), the variable (r) represents the radius of the circle. The equation itself is derived from the definition of a circle in a coordinate plane, where ((h, k)) is the center of the circle, and (r) is the distance from the center to any point on the circle.

By squaring the radius ((r)), the equation shows the relationship between the coordinates of points on the circle and the fixed distance (r) from the center. Therefore, the value of (r) is directly tied to the measurement of the radius, making it critical to understand that whenever someone refers to (r) in this context, they are referring specifically to the radius, not to other properties of the circle such as diameter, circumference, or area. The diameter is twice the radius, the circumference is the distance around the circle based on the radius, and the area is the space enclosed within the circle, all of which stem from the radius but are distinctly different concepts. Thus, the correct interpretation of (r) in this formula is that it is the radius of the circle.