Tangent of a Circle Calculator: Complete Guide to Tangent Lines (2026)

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. The tangent line is always perpendicular to the radius drawn to the point of tangency, creating a right angle that connects circles to the Pythagorean theorem. Whether you are designing gear systems, calculating light ray reflections, or solving a geometry problem, understanding how to find the tangent line from an external point to a circle gives you access to critical geometric relationships. The formula t = √(d² - r²) provides the tangent length directly from the circle radius and the distance to the external point.

What Is a Tangent to a Circle?

A tangent to a circle is a line that intersects the circle at exactly one point. Unlike a secant, which crosses through the circle at two points, the tangent merely touches the circle\'s edge. This single point of contact is called the point of tangency.

The most important property of a tangent line is that it is perpendicular to the radius at the point of tangency. This creates a right angle, which is why the Pythagorean theorem applies to tangent calculations. The triangle formed by the external point, the center, and the point of tangency is always a right triangle.

The Formula: Tangent Length from an External Point

The formula comes directly from the Pythagorean theorem. Since the tangent is perpendicular to the radius at the point of tangency, the triangle formed by the external point (P), the center (O), and the point of tangency (T) is a right triangle with the right angle at T.

Applying the Pythagorean theorem: PO² = PT² + OT², which gives d² = t² + r². Solving for t: t = √(d² - r²).

How to Use the Tangent of a Circle Calculator

1. Enter the circle radius.
2. Enter the distance from the external point to the center of the circle.
3. The calculator shows the tangent length and the angle at the external point.

Worked Examples

Notice that each example uses a Pythagorean triple (3-4-5, 5-12-13, 7-24-25), which makes the calculations clean. For non-triple values, the calculator handles the square root automatically.

Common Mistakes to Avoid

• Using d² + r² instead of d² - r² — the tangent is the hypotenuse\'s complement, not the hypotenuse.
• Trying to draw a tangent from a point inside the circle — it is impossible.
• Forgetting that two tangent lines of equal length can always be drawn from an external point.

Frequently Asked Questions

What is the formula for tangent length?

t = √(d² - r²), where d is the distance from the external point to the center and r is the radius.

Why is the tangent perpendicular to the radius?

This is a fundamental theorem: the tangent touches the circle at one point, and the radius to that point is perpendicular to the tangent line.

Can I draw a tangent from inside the circle?

No. Tangent lines can only be drawn from points on or outside the circle. From inside, no tangent exists.

How many tangent lines can be drawn from an external point?

Exactly two, both of the same length, symmetric about the line from the point to the center.

Conclusion

The tangent line is a fundamental concept in circle geometry with applications in engineering, optics, and computer graphics. Use our free tangent of a circle calculator to find the tangent length, angle, and point of tangency from any external point to any circle.