Calculate the tangent line from an external point to a circle. Find tangent length, tangent points, slopes, angles between tangent lines, and the power of a point.
A tangent line to a circle is a straight line that touches the circle at exactly one point. At that point, the tangent is perpendicular to the radius — a fundamental property that underlies countless constructions in geometry, engineering, and computer graphics.
From any point outside a circle, exactly two tangent lines can be drawn. These tangent lines have equal length (measured from the external point to the touch point), and they form a symmetric pair around the line connecting the external point to the circle's center. The tangent length T satisfies the elegant relationship T² + r² = d², where r is the radius and d is the distance from the point to the center. This is just the Pythagorean theorem applied to the right triangle formed by the center, the tangent point, and the external point.
The angle between the two tangent lines depends on how far the external point is from the circle. The closer the point, the wider the angle; at infinity, the two tangent lines become parallel. The quantity d² − r² is called the "power of the point" with respect to the circle — a key concept in inversive geometry.
This calculator supports two modes: (1) finding both tangent lines from an external point, and (2) finding the tangent line at a specific point on the circle. In both modes, enter the circle's center, radius, and the point of interest. The calculator returns tangent lengths, tangent points, slopes, line equations, and angles. Presets cover classic configurations like the 5-12-13 triangle and the unit circle. A reference table summarizes all key tangent-line formulas.
This calculator is useful when you need more than a single tangent length formula. In coordinate geometry problems, you often need the actual tangent points, the slope of each tangent line, the angle between the tangents, and confirmation that the point lies outside the circle. Doing all of that manually means switching between the distance formula, right-triangle relationships, and line equations.
It is especially helpful for analytic-geometry homework, drafting and CAD sketches, wheel and pulley layouts, and proof checking. The dual modes also let you move between two common tasks: drawing both tangents from an external point, or finding the unique tangent at a known point on the circle.
Tangent length: T = √(d² − r²), where d = distance from point to center Half angle: α = arcsin(r / d) Angle between two tangents: 2α Power of a point: P = d² − r² Tangent at point on circle: slope = −(x − cx) / (y − cy) (perpendicular to radius) Line equation: (x−cx)(px−cx) + (y−cy)(py−cy) = r²
Result: Tangent length = 12, angle between tangents ≈ 45.24°, half angle ≈ 22.62°
The circle has center (0,0) and radius 5. The external point is at (13,0), so d = 13. Tangent length = √(169 − 25) = √144 = 12. Half angle α = arcsin(5/13) ≈ 22.62°. The two tangent lines form a 45.24° angle at the external point. This is the classic 5-12-13 Pythagorean triple.
The key picture is a right triangle formed by the circle center, the tangent point, and the external point. Because a radius to a tangent point is always perpendicular to the tangent line, the tangent segment becomes one leg of the triangle, the radius is the other leg, and the center-to-point distance is the hypotenuse. That is why the relationship T² = d² − r² appears so naturally. The calculator exposes that relationship directly through the distance bar chart so you can see how the radius and tangent segment combine.
These are related but different tasks. From an external point, there are two valid tangents and two touch points, which is why the calculator returns paired slopes and coordinates. At a point already on the circle, there is only one tangent direction, determined by the fact that it must be perpendicular to the radius there. Switching modes helps you see the difference between constructing tangents and analyzing a known point of contact.
Most mistakes come from using the wrong point classification. If the point is inside the circle, no real tangent exists. If it is exactly on the circle, the tangent length from that point collapses to zero and the problem changes form. Another common issue is confusing the tangent slope with the radius slope; they are negative reciprocal directions when both are defined. Use the calculator to verify these cases before you simplify equations by hand.
A tangent line is a straight line that touches a circle at exactly one point. At that point, the tangent is perpendicular (90°) to the radius drawn to the same point.
Use T = √(d² − r²), where d is the distance from the external point to the circle's center and r is the radius. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
Exactly two. They are symmetric about the line from the external point to the center, and both have the same length.
The power of a point P with respect to a circle is d² − r², where d is the distance from P to the center. It equals the square of the tangent length and has important properties in projective geometry.
No tangent line can be drawn from a point inside the circle — the point must be outside (d > r) or on the circle (d = r). The calculator will show no results for interior points.
The calculator computes both tangent points. They lie on the circle at the positions where the tangent length meets the circle, forming right angles with the radius at each contact point.