Tangent to a Circle Calculator — Length & Angle
Given a circle of radius r and an external point at distance d from the center, find the length of the tangent line, the angles it makes, and the power of the point.
Distance from the external point to the tangent point on the circle: L = √(d² − r²)
- 1
d² − r²
13² − 5² = 169 − 25 = 144By the Pythagorean theorem: the radius and tangent length form a right angle, so L² = d² − r². - 2
Tangent length L = √(d² − r²)
√144 = 12
How does this calculator work?
Given circle radius r and external-point distance d (d > r), the tangent length is L = √(d² − r²). The tangent line meets the radius at a right angle. The angle at the external point is arccos(r/d). Example: r = 5, d = 13 → L = √(169 − 25) = √144 = 12. Power of the point = L² = 144.
Formula
How this is calculated
When a line from an external point is tangent to a circle, it touches the circle at exactly one point. At that tangent point, the radius is perpendicular to the tangent line — a fundamental property of tangent lines. This creates a right triangle with the hypotenuse being the distance d from the external point to the circle's center, one leg being the radius r, and the other leg being the tangent length L. By the Pythagorean theorem, L = √(d² − r²).
The angle α at the center (between the line to the external point and the radius to the tangent point) is arcsin(r/d). The angle β at the external point (between the tangent line and the line to the center) is arccos(r/d) — note α + β = 90° since the angle at the tangent point is always exactly 90°. From any external point exactly two tangent lines can be drawn, and they are symmetric about the line connecting the external point to the center.
The "power of a point" P = d² − r² = L² is a projective geometry concept: it is constant for all chords and tangents through that external point. It equals the square of the tangent length and is positive for external points, zero when the point lies on the circle, and negative for interior points. The triangle formed by the center, the external point, and the tangent point has area ½ × r × L.
Frequently asked questions
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. At that point, the radius drawn to the point of tangency is always perpendicular (at 90°) to the tangent line — this is the key property used to derive the tangent length formula.
Yes — from any point outside a circle, exactly two tangent lines can be drawn, one on each side. Both tangent lines have equal length (L = √(d² − r²)), and they are symmetric about the line from the external point to the center. This calculator gives the length of one (and both) of those tangents.
A point inside a circle cannot be connected to the circle by a tangent line — no such tangent exists. d² − r² would be negative, so √(d² − r²) is not a real number. The calculator requires d > r and shows a warning if that condition is not met.
Also known as
TG we-Calculate Editorial Team. (2026). Tangent to a Circle Calculator — Length & Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/tangent-circle-calculator
TG we-Calculate Editorial Team. "Tangent to a Circle Calculator — Length & Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/tangent-circle-calculator.
TG we-Calculate Editorial Team, "Tangent to a Circle Calculator — Length & Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/tangent-circle-calculator
@misc{wecalculate_tangent_circle_calculator, title = {Tangent to a Circle Calculator — Length & Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/tangent-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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