Tangent to a Circle

Q: What is a tangent of a circle?

A tangent of a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at that point. Unlike a secant, which intersects the circle at two points, a tangent only touches the circle at a single point of tangency.

Q: How do you calculate the length of a tangent to a circle?

The length of a tangent from an external point to a circle is calculated using the Pythagorean theorem: l = sqrt(d^2 - r^2), where l is the tangent length, d is the distance from the center to the external point, and r is the radius of the circle. The tangent, radius, and the line connecting the center to the external point form a right triangle.

Q: What if the external point is inside the circle?

If the distance from the center to the external point is less than or equal to the radius (d r) for a valid tangent.

Find the tangent length from any external point to a circle using l = sqrt(d^2 - r^2). Free online geometry calculator with instant results for students and engineers.

Find tangent length

Radius (units):

Distance from center to external point (units):

About This Calculator

The Tangent to a Circle Calculator computes the length of the tangent line segment from an external point to a circle. A tangent is a line that touches a circle at exactly one point -- the point of tangency -- and is perpendicular to the radius drawn to that point. This fundamental geometric concept is widely used in geometry, engineering, physics, and calculus.

The tangent length is calculated using the Pythagorean theorem. When you draw a tangent from an external point T to a circle with center O and radius r, the radius OA (where A is the point of tangency) is perpendicular to the tangent line AT. Together with the line OT (the distance from the center to the external point), these three segments form a right triangle with OT as the hypotenuse. The formula for tangent length is l = sqrt(d^2 - r^2), where d is the distance from the center to the external point and r is the radius of the circle.

For example, a circle with radius 10 units and an external point 15 units from the center gives a tangent length of sqrt(15^2 - 10^2) = sqrt125 ≈ 11.18 units. If the external point is inside or on the circle (d <= r), a real tangent does not exist since the tangent line would not reach the circle from an interior point.

Applications of Circle Tangents

Tangents to circles appear in numerous real-world contexts: belt and pulley systems in mechanical engineering use tangent lengths to determine belt sizes; tangential polygons and incircles are fundamental in geometric constructions; the slope of a tangent line to a curve gives instantaneous rates of change in calculus; and tangent lines are used in computer graphics for collision detection and rendering.

Regional Notes

This geometry calculator is unit-agnostic -- it works with any measurement system (metric, imperial, or arbitrary units). Students and professionals in the United States (US customary), United Kingdom (imperial), and India (metric) can all use the same calculator by entering their values in consistent units. The result is in the same unit as the input values.

Frequently Asked Questions

What is a tangent of a circle?

A tangent of a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at that point. Unlike a secant, which intersects the circle at two points, a tangent only touches the circle at a single point of tangency.

How do you calculate the length of a tangent to a circle?

The length of a tangent from an external point to a circle is calculated using the Pythagorean theorem: l = sqrt(d^2 - r^2), where l is the tangent length, d is the distance from the center to the external point, and r is the radius of the circle. The tangent, radius, and the line connecting the center to the external point form a right triangle.

What if the external point is inside the circle?

If the distance from the center to the external point is less than or equal to the radius (d <= r), a real tangent line does not exist because the point lies inside or on the circle. The calculator will show N/A for the tangent length in this case. The external point must be outside the circle (d > r) for a valid tangent.

What are some applications of circle tangents?

Circle tangents have many applications including: calculating belt lengths in pulley systems, constructing tangential polygons and incircles in geometry, finding instantaneous velocity in mechanics through the slope of a tangent curve, and solving differential approximation problems in calculus.

What is the equation of a tangent to a circle?

The equation of the tangent line to a circle with center (a,b) and radius r at a point (x1,y1) on the circle is given by: (x - x1)(x1 - a) + (y - y1)(y1 - b) = 0. Since the tangent is perpendicular to the radius at the point of tangency, this linear equation represents the unique tangent line at that point.

Can the tangent length be negative?

No, the tangent length is always a non-negative value. When the external point is outside the circle (d > r), the tangent length is positive. When the point lies on the circle (d = r), the tangent length is zero. There is no real-valued tangent length when the point is inside the circle.

Is this tangent calculator free to use?

Yes, this tangent of a circle calculator is completely free to use with no registration or login required. You can calculate as many tangent lengths as you need for geometry problems, engineering projects, or educational purposes.