For AI agents: use /llms.txt for the Nakafa content index.
Given a circle and a point, exactly one of three cases applies: the point lies inside the circle, on its circumference, or outside it.
The same comparison appears in simplified coverage models, such as deciding whether a location lies inside an idealized circular signal or radar range. The circle equation and point coordinates then classify the location exactly.
To classify the point, we use its power with respect to the circle. This algebraic quantity can be positive, zero, or negative. An ordinary distance is always zero or positive. For a circle with center and radius :
For point and the general circle , the power of is:
Substituting the coordinates once gives a value whose sign determines the point's location.
Based on the power of a point value, there are three possible positions:
The point is inside the circle when .
The point is on the circle when .
The point is outside the circle when .
The three cases are shown in the graph below:
First identify the circle equation and the point coordinates. Substitute the coordinates into the power-of-a-point formula to obtain .
The sign of determines the point's position. If , the point is inside the circle. If , the point is on the circle. If , the point is outside the circle.
For point and circle , first compute the squared distance to the center. Then subtract :
Because , point lies inside the circle.
For , comparing with is equivalent to checking the sign of .
For a circle in general form , calculate . The sign of determines whether the point is inside, on, or outside the circle.
For example, for point and circle :
Since , then point is located inside the circle.
Determine the position of point relative to circle .
Investigate whether point lies on circle .
Determine the position of point relative to circle .
A circle has equation . Determine the position of point relative to that circle.
Solution:
Substitute coordinates of point into circle equation :
Since , point lies outside the circle.
Solution:
Substitute coordinates of point into circle equation :
Solution:
Substitute coordinates of point into circle equation :
Solution:
For circle with center and radius .
Published: . Updated: .
Since , point lies exactly on the circle.
Since , then point is located inside the circle.
Substitute point :
Since , point lies outside the circle.
