Tangent of a Circle Calculator

Enter a circle's center and radius plus an external point to find the tangent length, the distance to the center, and the angle between the two tangent lines.

Quick Facts

Tangent length
L = √(d² − r²)
d is the distance from the external point to the center; r is the radius.
Tangent ⟂ radius
90° at the point of tangency
A tangent line touches the circle at exactly one point and is always perpendicular to the radius there.
Angle between tangents
θ = 2·arcsin(r/d)
The angle formed at the external point by the two tangent lines.
Equal tangent segments
Both tangents share length L
The two tangent segments drawn from one external point to a circle are always equal.

Your Results

Calculated
Distance to Center (d)
-
d = √((x−h)² + (y−k)²)
Tangent Length (L)
-
L = √(d² − r²)
Angle Between Tangents
-
θ = 2·arcsin(r/d)
Point Position
-
Relative to the circle

Ready

Enter the circle's center, radius, and an external point, then press Calculate.

Formula and Method for the Tangent to a Circle

A tangent line to a circle touches the circle at exactly one point — the point of tangency — and never crosses into the circle's interior. Given a circle with center (h, k) and radius r, and an external point (x₁, y₁) outside the circle, this calculator finds the length of the tangent segment from that point to the circle, the straight-line distance to the center, and the angle formed between the two possible tangent lines.

How the calculation works

The center, the external point, and either point of tangency form a right triangle. The radius drawn to the point of tangency is always perpendicular to the tangent line, so that triangle has a right angle at the point of tangency. First find the distance from the external point to the center: d = √((x₁−h)² + (y₁−k)²). That distance is the hypotenuse of the right triangle, with legs r (the radius) and L (the tangent length). By the Pythagorean theorem, r² + L² = d², so the tangent length is L = √(d² − r²). The angle between the two tangent lines, measured at the external point, is θ = 2·arcsin(r/d), since sin of the half-angle equals the opposite leg (r) over the hypotenuse (d).

Common mistakes

  • Point inside the circle: if d < r, the point is inside the circle and no real tangent line exists — d² − r² would be negative under the square root.
  • Confusing d with L: d is the straight-line distance to the center; L is the shorter tangent segment length. They are only equal when the radius is zero, which never happens for a real circle.
  • Degrees vs. radians: arcsin returns radians in most programming languages and calculators — convert to degrees (multiply by 180/π) before comparing to a protractor reading.

Real-world applications

  • Line-of-sight and horizon calculations treat the Earth as a circle and the observer's height as defining the tangent length to the visible horizon.
  • Belt and pulley design uses tangent lines to find the straight sections of belt that connect two circular pulleys.
  • CAD and technical drawing rely on tangent constructions to blend straight edges smoothly into circular arcs and fillets.
  • Surveying and navigation use tangent-length calculations to determine sightlines around circular obstacles or boundaries.

Frequently Asked Questions

What is the formula for the tangent length from a point to a circle?
The tangent length is L = √(d² − r²), where d is the distance from the external point to the circle's center and r is the radius. This comes from the Pythagorean theorem applied to the right triangle formed by the center, the external point, and the point of tangency.
Why are the two tangent segments from the same external point always equal?
Both tangent segments form right triangles that share the same hypotenuse (the segment from the external point to the center) and have equal legs (the radius). By the hypotenuse-leg congruence theorem those triangles are congruent, so the tangent segments are equal in length.
What happens if the point is inside the circle?
No tangent line exists from a point inside the circle, because every line through it crosses the circle at two points instead of touching it at one. The calculator flags this as invalid input.
How do I find the angle between the two tangent lines?
Use θ = 2·arcsin(r/d), where r is the radius and d is the distance from the external point to the center. This doubles the half-angle formed between the line to the center and each tangent line.

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