Tangent of a Circle Calculator
Calculate tangent line length from an external point to a circle using geometry formulas
Calculate Tangent Properties
Radius of the circle
Distance from circle center to external point
Example Calculation
Find Tangent Length
Given: Circle radius r = 10 m, External point distance d = 15 m
Find: Length of tangent from external point to circle
Formula: l = √(d² - r²)
Solution: l = √(15² - 10²) = √(225 - 100) = √125 = 11.18 m
Key Properties
• Tangent line is perpendicular to radius at point of contact
• Forms a right triangle with radius and distance to center
• Two tangent lines from external point have equal length
• External point must be outside the circle (d > r)
Tangent Formulas
Tangent Properties
Perpendicular to radius at contact point
Equal length tangents from external point
Forms right triangle with radius
External point must be outside circle
Quick Tips
For internal points (d < r), no real tangent exists
When d = r, point is on circle (tangent length = 0)
Two tangent lines can be drawn from any external point
Used in engineering, optics, and geometric constructions
Understanding Circle Tangents
What is a Circle Tangent?
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 perpendicular to the radius drawn to the point of contact.
Geometric Properties
- •Tangent is perpendicular to radius at point of contact
- •Two tangent lines from external point have equal length
- •Tangent line never intersects the circle at two points
- •Angle between two tangents = 2 × angle at center
Mathematical Foundation
Right Triangle Formation:
• O = Center of circle
• T = External point
• A = Point of tangency
• Triangle OTA is a right triangle
• ∠OAT = 90° (tangent ⊥ radius)
Pythagorean Theorem Application
Since the tangent forms a right triangle with the radius and the line from center to external point, we can use: r² + l² = d²
Engineering Applications
Used in belt drive systems, cam design, and mechanical linkages where smooth tangential motion is required.
Geometric Construction
Essential for drawing tangent polygons, inscribed shapes, and complex geometric patterns.
Calculus Connection
In coordinate geometry, tangent lines represent instantaneous rate of change and are fundamental to differential calculus.