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Tangent Line to Circle Calculator

MathTriangles and Trigonometry
Circle
Point
Tangent length from the point —
Point position
—
Tangent line 1
—
Tangent line 2
—
Point(s) of tangency
—
Distance from centre to point
—
Angle between tangents (degrees)
—

A tangent to a circle is perpendicular to the radius at the point where it touches. From an outside point at distance d from the centre, the tangent length is √(d² − r²). For a circle of radius 5 centred at the origin and the point (13, 0), that is √(169 − 25) = 12, and the two tangents are y = ±0.4167x ∓ 5.4167.

About this tool

Analytic geometry homework, CAD sketches and machining layouts all ask the same question: which straight lines through a given point just touch a given circle? Enter the centre (h, k), the radius and the point. If the point lies on the circle you get its single tangent; if it lies outside you get both tangent lines in slope-intercept form (or x = c when a tangent is vertical), the two points of tangency, the tangent segment length and the angle the tangents make at the point. The working shows the distance, the Pythagorean step and the tangency points. A point inside the circle has no real tangent, and the tool says so instead of returning a line. Coordinates are shown to four decimals, so a slope like -5/12 appears as -0.4167.

How to use it

  1. Describe the circle

    Type the centre coordinates h and k and the radius r, all in the same length unit.

  2. Enter the point

    Give the x and y of the point the tangent lines must pass through.

  3. Read the tangents

    Both line equations, the tangency points, the tangent length and the angle between the lines update as you type, with the working underneath.

Examples

Point (3, 4) on x² + y² = 25

Centre x (h)
0
Centre y (k)
0
Radius (r)
5
Point x
3
Point y
4

Result Tangent length from the point: 0
Point position: On the circle (one tangent)
Tangent line 1: y = -0.75x + 6.25
Tangent line 2: —
Point(s) of tangency: (3, 4)
Distance from centre to point: 5
Angle between tangents (degrees): 180

  1. Circle: (x − 0)² + (y − 0)² = 25
  2. Distance from centre to point d = √(3² + 4²) = 5
  3. d = r, so the point lies on the circle and there is one tangent, perpendicular to the radius.
  4. Tangent: (3)(x − 3) + (4)(y − 4) = 0 → y = -0.75x + 6.25

The radius to (3, 4) has slope 4/3, so the tangent has slope -3/4 and meets the y-axis at 25/4 = 6.25.

Point (13, 0), radius 5

Centre x (h)
0
Centre y (k)
0
Radius (r)
5
Point x
13
Point y
0

Result Tangent length from the point: 12
Point position: Outside the circle (two tangents)
Tangent line 1: y = -0.4167x + 5.4167
Tangent line 2: y = 0.4167x − 5.4167
Point(s) of tangency: (1.9231, 4.6154) and (1.9231, -4.6154)
Distance from centre to point: 13
Angle between tangents (degrees): 45.24

  1. Circle: (x − 0)² + (y − 0)² = 25
  2. Distance from centre to point d = √(13² + 0²) = 13
  3. Tangent length = √(d² − r²) = √(169 − 25) = 12
  4. Tangent points T = C + (r²/d²)·v ± (r·L/d²)·v⊥ = (1.9231, 4.6154) and (1.9231, -4.6154)
  5. Each tangent is perpendicular to its radius CT: y = -0.4167x + 5.4167 and y = 0.4167x − 5.4167
  6. Angle between the tangents = 2·asin(r/d) = 45.2397°

The radius, the tangent segment and the line to the centre form a 5-12-13 right triangle, so each tangent segment is 12 long.

Circle centred at (2, 3), point (8, 11)

Centre x (h)
2
Centre y (k)
3
Radius (r)
5
Point x
8
Point y
11

Result Tangent length from the point: 8.6603
Point position: Outside the circle (two tangents)
Tangent line 1: y = 0.4272x + 7.5827
Tangent line 2: y = 8.3001x − 55.4009
Point(s) of tangency: (0.0359, 7.5981) and (6.9641, 2.4019)
Distance from centre to point: 10
Angle between tangents (degrees): 60

  1. Circle: (x − 2)² + (y − 3)² = 25
  2. Distance from centre to point d = √(6² + 8²) = 10
  3. Tangent length = √(d² − r²) = √(100 − 25) = 8.6603
  4. Tangent points T = C + (r²/d²)·v ± (r·L/d²)·v⊥ = (0.0359, 7.5981) and (6.9641, 2.4019)
  5. Each tangent is perpendicular to its radius CT: y = 0.4272x + 7.5827 and y = 8.3001x − 55.4009
  6. Angle between the tangents = 2·asin(r/d) = 60°

The point is 10 units from the centre, twice the radius, so the tangent length is √75 and the two tangents meet at 60 degrees.

How it is calculated

L = √(d² − r²); T = C + (r²/d²)·v ± (r·L/d²)·v⊥

C = (h, k)
centre of the circle
r
radius
v
vector from the centre to the point, (x − h, y − k)
d
length of v, the centre-to-point distance
L
length of each tangent segment
T
a point of tangency; v⊥ is v turned 90 degrees

Because the radius CT meets the tangent at a right angle, the triangle formed by the centre, the point and T is right-angled at T, giving L = √(d² − r²). The foot of T along v sits at r²/d of the way out, and it is offset sideways by r·L/d, which yields the two tangency points. Each tangent is then the line through T whose normal is CT: (T − C)·(X − T) = 0. The angle between the two tangents is 2·asin(r/d).

When not to use it

  • Tangents to an ellipse or parabola need a different condition than perpendicularity to a radius.
  • Common tangents shared by two circles are a separate construction.
  • A point inside the circle has no real tangent line at all.

Common mistakes

  • Entering r² from the equation as the radius instead of its square root.
  • Forgetting the sign flip in (x − h): the circle (x + 2)² means h = −2.
  • Using the radius slope as the tangent slope rather than its negative reciprocal.

Frequently asked questions

Why is a tangent perpendicular to the radius?

The point of tangency is the closest point on the line to the centre. The shortest segment from a point to a line is the perpendicular one, so the radius drawn to that point meets the tangent at 90 degrees.

How many tangents pass through a point?

Two if the point is outside the circle, exactly one if it lies on the circle, and none if it is inside. The tool reports which case applies under Point position.

What is the tangent at a point on x² + y² = r²?

For a circle centred at the origin the tangent at (x1, y1) is x·x1 + y·y1 = r². At (3, 4) on x² + y² = 25 that is 3x + 4y = 25, or y = -0.75x + 6.25.

Why does a line show as x = c?

A vertical tangent has no finite slope, so it cannot be written as y = mx + b. It happens when the tangency point sits level with the centre, directly left or right of it.

How is the tangent length related to the power of a point?

The square of the tangent length, d² − r², is the power of the point with respect to the circle. It also equals the product of the two segments of any secant drawn from the same point.

What does the angle between tangents tell me?

It is the angle at the outside point between the two tangent lines, 2·asin(r/d). When the point is twice the radius from the centre, r/d is 0.5 and the angle is 60 degrees; it shrinks towards zero as the point moves away.

Cite this tool

Using this tool in an assignment, article or report? Copy a citation, or a link for your website.

APA

Tools Directory Editorial Team. (2026, October 13). Tangent Line to Circle Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/triangles-and-trigonometry/tangent-line-to-circle-calculator/

MLA

Tools Directory Editorial Team. “Tangent Line to Circle Calculator.” Tools Directory, 13 Oct. 2026, https://tools.milunir.com/math/triangles-and-trigonometry/tangent-line-to-circle-calculator/. Accessed 26 Oct. 2026.

Chicago

Tools Directory Editorial Team. “Tangent Line to Circle Calculator.” Tools Directory. Last modified October 13, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/triangles-and-trigonometry/tangent-line-to-circle-calculator/.

Link<a href="https://tools.milunir.com/math/triangles-and-trigonometry/tangent-line-to-circle-calculator/">Tangent Line to Circle Calculator – Tools Directory</a>