What is a tangent to a circle?

A tangent is a straight line that touches the circle at exactly one point, called the tangent point. At this point, the tangent is perpendicular to the radius.

How do I find the tangent length from an external point?

Use the formula: tangent length = √(d² − r²), where d is the distance from the external point to the center and r is the radius.

What is the power of a point?

The power of a point P with respect to a circle is d² − r². It equals the square of the tangent length when P is outside the circle.

Why are both tangent lengths equal?

By the symmetry of the configuration (two right triangles sharing the hypotenuse PC and having equal legs r), the two tangent segments are congruent.

Can I find tangent lines if the point is inside the circle?

No. If the point is inside the circle (d < r), no real tangent line exists from that point. The calculator will show an error.

What angle do the two tangent lines form?

The angle between the two tangent lines at the external point equals 2 × arcsin(r/d). It approaches 180° as the point moves toward the circle and approaches 0° as the point moves far away.

Tangent to a Circle Calculator

Calculate tangent line properties from an external point to a circle. Find tangent length, tangent points, angles, and power of a point. Includes ratio bars, 8 presets, and a reference table.

Presets:

Circle Center X

Circle Center Y

Circle Radius

External Point X

External Point Y

Angle Unit

Decimal Precision

Tangent Length

12.0000

Distance from external point to tangent point

Distance to Center

13.0000

Distance from external point to circle center

Tangent Point 1

(1.9231, 4.6154)

First point where tangent touches the circle

Tangent Point 2

(1.9231, -4.6154)

Second point where tangent touches the circle

Angle Between Tangents

45.2397°

Angle formed at the external point by the two tangent lines

Central Angle

134.7603°

Angle subtended at the center by the two tangent points

Half Angle

67.3801°

Angle between center-line and each tangent (sin⁻¹(r/d))

Power of the Point

144.0000

d² − r² = tangent length squared

Geometry Ratios

r / d (radius to distance): 0.3846

Tangent / Distance: 0.9231

Tangent / Radius: 2.4000

Reference: Tangent Properties for Common Configurations

Radius	Distance	Tangent Length	Angle Between Tangents
1	5	4.8990	156.93°
3	5	4.0000	106.26°
5	13	12.0000	134.76°
7	25	24.0000	147.48°
10	26	24.0000	134.76°
2	5	4.5826	132.84°

Planning notes, formulas, and examples

About the Tangent to a Circle Calculator

A tangent line to a circle is a line that touches the circle at exactly one point — the tangent point — and is perpendicular to the radius at that point. From any external point outside a circle, exactly two tangent lines can be drawn, and both tangent segments have equal length. This property is central to Euclidean geometry and has practical applications in engineering (cam profiles, belt-and-pulley systems), computer graphics (smooth curve joining), and navigation. The <strong>tangent length</strong> from an external point P to a circle with center C and radius r is given by √(d² − r²), where d is the distance from P to C. The <strong>power of the point</strong> with respect to the circle is d² − r², and equals the square of the tangent length. This calculator lets you enter the circle center (cx, cy), the radius, and an external point (px, py). It computes the tangent length, both tangent points (the coordinates where each tangent touches the circle), the angle between the two tangent lines at the external point, the central angle subtended by the tangent points, the half-angle, and the power of the point. Visual ratio bars show the proportions of key geometric quantities. A reference table covers several standard configurations for quick verification. Eight presets let you explore classic setups like the unit circle or Pythagorean-triple configurations.

When This Page Helps

This calculator is useful when a tangent problem involves coordinates rather than a simple diagram. Once the center, radius, and external point are known, you often need more than the tangent length alone: the actual tangent-point coordinates, the angle between the two tangent lines, and the power of the point are all tied together. Doing that by hand means combining distance formulas, inverse trig, and coordinate geometry in one chain of work.

It is especially helpful for verification in analytic geometry, CAD-style layouts, pulley or cam sketches, and classroom proofs. Because both tangent points are returned explicitly, you can check whether a line touches the circle at the correct place instead of relying only on a length calculation.

How to Use the Inputs

Enter the circle center coordinates (X and Y).
Enter the circle radius.
Enter the external point coordinates (X and Y).
Choose whether to display angles in degrees or radians.
Optionally adjust decimal precision.
Click a preset to load a standard configuration.
Read tangent length, tangent points, angles, and more.

Formula used

Tangent Length = √(d² − r²)
Power of Point = d² − r²
Half Angle α = arccos(r / d)
Angle Between Tangents = 2α
Tangent Points: C + r·[cos(θ ± α), sin(θ ± α)] where θ = atan2(Py−Cy, Px−Cx)

Example Calculation

Result: Tangent Length = 12, Power of the Point = 144, Tangent Points ≈ (1.9231, 4.6154) and (1.9231, -4.6154)

Enter the circle center at (0, 0), radius 5, and external point (13, 0). The distance from the point to the center is 13, so the tangent length is √(13² − 5²) = √144 = 12. The power of the point is the same value squared, 144. Using the tangent-point formulas with θ = 0 and α = arccos(5/13), the contact points are approximately (1.9231, 4.6154) and (1.9231, -4.6154). The calculator also reports the angle between the tangents and the corresponding central angle automatically.

Tips & Best Practices

Both tangent segments from the same external point are always equal in length — a useful symmetry check.
The tangent length equals zero when the point lies on the circle (the "tangent" is just one point).
Power of a point is negative when the point is inside the circle — no real tangent exists.
In 3D, tangent lines to a sphere from an external point form a cone — the same formulas apply in the cross-section.

Equal Tangents From an External Point

One of the most important circle theorems says that the two tangent segments drawn from the same external point are equal. This is not just a length fact; it reflects the symmetry of the two right triangles formed by the center, the external point, and each tangent point. In coordinate problems, that symmetry gives you a reliable check: if the computed tangent lengths do not match, something in the setup is wrong.

Power of a Point and Why It Helps

The expression d² − r² appears throughout tangent and secant geometry. For an external point, it equals the square of the tangent length, which is why it is called the power of the point. This value is useful because it tells you immediately whether a real tangent exists. If the point lies outside the circle, the power is positive. On the circle, it is zero. Inside the circle, it is negative and no real tangent line can be drawn.

Reading the Angles in the Diagram

The angle between the two tangent lines gets smaller as the external point moves farther away, because the circle looks narrower from that location. The central angle subtended by the tangent points complements that view from the center. Looking at both angles together helps you connect the coordinate output to the actual geometry, especially when you are sketching the configuration or checking a proof involving arcs, radii, and perpendicular tangents.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

A tangent is a straight line that touches the circle at exactly one point, called the tangent point. At this point, the tangent is perpendicular to the radius.