How to use the circle tangent and intersection calculator
- Choose Tangents from a point, Two circles or Line and circle.
- Enter the center and radius of the circle, then the point, the second circle or two points on the line.
- Read the coordinates of each tangent point or intersection and the lengths shown with them.
- Use the steps to see which formula gave each answer and check it against your drawing.
Formula
Tangent length from a point at distance d: sqrt(d^2 - r^2)
Two circles: a = (d^2 + r1^2 - r2^2) / (2d), h = sqrt(r1^2 - a^2)
Outer common tangent: sqrt(d^2 - (r1 - r2)^2); inner: sqrt(d^2 - (r1 + r2)^2)
Line and circle: distance = |cross product| / length, chord = 2 sqrt(r^2 - distance^2)
d is the distance between centers, or from the point to the center. The tangent point is found a right angle away from the radius, using cos(angle) = r / d at the center.
Three small cases
A circle of radius 5 at the origin and the point (13, 0): the distance is 13, so the tangent length is sqrt(169 - 25) = 12. The two tangent points are (1.923077, 4.615385) and (1.923077, -4.615385), the tangent lines meet at an angle of 45.2397 degrees, and the chord joining the points of contact is 9.230769 long.
Two circles of radius 5 centered at (0, 0) and (8, 0) cross at (4, 3) and (4, -3), have 2 common tangent lines, and each outer tangent segment is 8 long. The line through (-6, 3) and (6, 3) is 3 from the center of the radius 5 circle, cuts it at (-4, 3) and (4, 3), and the chord is 8 long.
How many common tangents two circles have
The count depends on how far apart the circles are. Apart from each other there are 4 common tangents: 2 outer and 2 crossing between them. When the circles touch from outside there are 3, and when they cross at two points there are 2. A circle touching another from the inside has 1, and a circle lying completely inside another has none. The page names the case before listing the points.
The circle calculator covers area and circumference, and the circle from three points calculator finds the circle through three corners. To measure how much two circles overlap use the circle overlap area calculator.
Points on the circle and edge cases
If your point lies exactly on the circle, there is a single tangent line and the point is its own tangent point. If it is inside the circle, no tangent line exists and the page asks for another point. If two circles share a center, they cannot meet unless they are identical, in which case every point is shared.
Assumptions and limits
- All coordinates and radii use one length unit and are treated as exact values.
- Circles lie in a flat plane. The line in the line mode is infinite, not only the segment between the two points.
- Touching is detected within a very small tolerance, so numbers like 7.0000000001 are treated as touching.
- Coordinates may be negative. Radii must be above zero.
Frequently asked questions
How do I find the tangent points from an outside point?
Measure the distance d from the point to the circle center. The tangent points sit at an angle of acos(r / d) either side of the line from the center to your point, at distance r from the center. The page does this for you and lists both.
What is the difference between an outer and an inner common tangent?
An outer common tangent touches both circles on the same side, without crossing between them. An inner common tangent crosses the line between the centers. Inner tangents exist only when the circles are apart.
Why does the calculator say the circles do not meet?
Either they are too far apart, so d is more than r1 + r2, or one is inside the other, so d is less than the difference of the radii. Compare the distance with those two values.
Is a line that touches the circle at one point a tangent?
Yes. When the distance from the center to the line equals the radius the line is a tangent and meets the circle at exactly one point.