This tangent of a circle calculator computes the length of the tangent of a circle. A tangent line of a circle is a very versatile property of a circle that will not change even during different transformations, mapping, and scalings. A tangent is very crucial to various geometric constructions and theorems.
A tangent can be used for different applications in the differentials and approximations, strength of materials, motion, distance, and so on. In this article, you'll read more on what is a tangent, and how to find the tangent of a circle, concluding with a discussion on several applications of tangents. In this tool, we use the distance formula.
What is a tangent of a circle?
The definition of the tangent of a circle is given as a line beginning from a point on a circle while being perpendicular to the radius. A tangent of the circle touches, not intersects, the circle at one point only.
Consider a circle with center O and a point A on the circle. The line joining the points O and A is OA. The tangent of a circle from the point A is perpendicular to the line OA. Now, draw a line between the point O and a point T on the tangent line.
Note that the tangent of a circle is perpendicular to the radius, i.e., the lines joining the points O, A, and T form a right-angled triangle. The equation of the tangent of a circle is therefore based on Pythagoras's theorem:
Let the radius of the circle be r, and the length of the tangent be l. The distance d between the external point T and the center of the circle O can also be written as:
In other words, the length of the tangent is given by the equation of the tangent to a circle:
How to find the tangent of a circle?
Consider a circle with center (a, b) and radius r. Its equation is:
If you know the coordinates of the external point (x, y), first find its distance to the center using the distance formula:
Then substitute the distance d and the radius r into the tangent formula l = √(d² − r²). For a real tangent to exist, the point must lie on or outside the circle, i.e., d ≥ r. If d = r, the point is on the circle and the tangent length is 0; if d < r, the point is inside and no tangent can be drawn.
How to use this calculator
- Choose a method — enter the distance d from the point to the center directly, or provide the coordinates of the center and the external point.
- Pick your unit system — metric (cm) or US/Imperial (in).
- Enter the radius of the circle.
- Enter the distance or the coordinates for the method you selected.
The calculator instantly returns the tangent length together with the distance to the center, the power of the point, the angle between the two tangents, the diameter, circumference, and area of the circle.
Example: Using the tangent of a circle calculator
Find the length of the tangent from a point d = 5 cm away from the center of a circle whose radius is r = 3 cm.
l = √(5² − 3²)
l = √(25 − 9) = √16 = 4 cm
The same result appears if you use the coordinates method with center (0, 0), radius 3 cm, and external point (4, 3): the distance is d = √(4² + 3²) = 5 cm, so the tangent length is again 4 cm.
Applications of the tangent line of a circle
- Differentials and approximations — a tangent line is the best linear approximation of a curve at a point, the foundation of linearization and Newton's method.
- Strength of materials — tangents and tangential stresses appear in the analysis of shafts, gears, and curved beams.
- Motion and distance — the velocity of an object moving along a circular path is always tangent to the circle at that point.
- Geometric constructions — tangents are used to draw belts and pulleys, road curves, and to solve locus problems.
- Optics — the law of reflection is stated with respect to the tangent (and normal) at the point of contact of a curved mirror.
Unit Systems
Metric System
Enter the dimensions in centimeters (cm). The tangent length is also shown in millimeters (mm), meters (m), and inches (in).
US / Imperial System
Enter the dimensions in inches (in). The tangent length is also shown in feet (ft), yards (yd), and centimeters (cm).
FAQs
What is a tangent of a circle?
A tangent of a circle is a straight line that touches the circle at exactly one point and is perpendicular to the radius drawn to that point of contact.
How do I find the length of a tangent from an external point?
Use l = √(d² − r²), where r is the radius of the circle and d is the distance from the external point to the center. This comes directly from the Pythagorean theorem applied to the right triangle formed by the radius, the tangent, and the line to the center.
Why is the tangent perpendicular to the radius?
Among all the segments from the center to points on the tangent line, the radius to the point of contact is the shortest. The shortest distance from a point to a line is always perpendicular to that line, so the radius meets the tangent at a right angle.
How many tangents can be drawn from a point?
It depends on where the point lies. From a point outside the circle you can draw exactly two tangents of equal length; from a point on the circle exactly one; and from a point inside the circle, none.
What is the power of a point?
The power of an external point is d² − r², which equals the square of the tangent length (l²). It is a useful invariant in circle geometry, appearing in the tangent–secant and intersecting-chords theorems.