📐 Tangent of a Circle Calculator
Calculate tangent properties from a point to a circle
Circle Information
External Point
How to Use This Calculator
Enter Circle Information
Input the center coordinates (h, k) and radius (r) of the circle.
Enter External Point
Input the coordinates of the point from which you want to draw tangents. This point should be outside the circle for tangents to exist.
Calculate
Click "Calculate Tangent" to get the tangent length, tangent point coordinates, and other properties.
Formulas
Tangent Length (from external point)
Tangent Length = √(d² - r²)
Where d = distance from point to circle center, r = radius
Key Properties:
- The tangent is always perpendicular to the radius at the point of contact (90° angle)
- From an external point, there are exactly two tangents to a circle
- Both tangents have equal length
- The tangent point lies on the circle and on the line from center to external point
Distance Check:
- If distance > radius: Point is outside, two tangents exist
- If distance = radius: Point is on circle, one tangent exists (at the point)
- If distance < radius: Point is inside, no real tangents exist
Example: Circle centered at (0,0) with r=5, external point at (10,0)
Distance from (10,0) to (0,0) = 10
Tangent length = √(10² - 5²) = √(100 - 25) = √75 ≈ 8.66 units
About Tangent of a Circle Calculator
The Tangent of a Circle Calculator finds properties of tangent lines drawn from an external point to a circle. A tangent is a line that touches the circle at exactly one point and is perpendicular to the radius at that point.
When to Use This Calculator
- Geometry Problems: Solve tangent line problems in coordinate geometry
- Engineering: Design systems with tangential connections
- Education: Learn and practice circle tangent concepts
- Navigation: Calculate paths tangent to circular boundaries
- Verification: Check manual tangent calculations
Why Use Our Calculator?
- ✅ Comprehensive Results: Shows tangent length, tangent point, and position analysis
- ✅ Position Detection: Determines if point is outside, on, or inside the circle
- ✅ Educational: Displays formulas and calculation steps
- ✅ Accurate: Precise mathematical calculations
- ✅ Instant Results: Calculate immediately
- ✅ 100% Free: No registration required
Understanding Tangents
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency. The fundamental property is that the radius to the point of tangency is perpendicular to the tangent line.
- From an external point, exactly two tangents can be drawn to a circle
- Both tangents have equal length
- The tangents are symmetric about the line from the external point to the circle's center
- The angle between the two tangents can be calculated using the tangent length and distance
Real-World Applications
Navigation: Calculate the shortest path that touches a circular boundary or obstacle.
Engineering: Design gears, pulleys, and mechanisms where components connect tangentially.
Physics: Analyze circular motion, orbits, and trajectories that are tangent to circular paths.
Frequently Asked Questions
How many tangents can be drawn from a point to a circle?
From an external point (outside the circle): exactly 2 tangents. From a point on the circle: exactly 1 tangent (at that point). From an interior point: 0 tangents (no real solutions).
Why is the tangent perpendicular to the radius?
This is a fundamental property of circles. At the point of tangency, the tangent line is perpendicular to the radius. This can be proven geometrically and is essential for many circle theorems.
What if the point is inside the circle?
If the point is inside the circle, no real tangents exist. In the complex plane, there would be imaginary tangents, but geometrically, you cannot draw a tangent from an interior point.
How do I find the equation of the tangent line?
You need the tangent point coordinates and the slope. The slope is perpendicular to the radius slope. If radius slope is m, tangent slope is -1/m (or undefined if m=0).
Can tangents be drawn to other shapes?
Yes! Tangents can be drawn to any smooth curve. For circles, tangents are particularly simple because of the constant radius property. For ellipses, parabolas, etc., the tangent formulas are different.
What's the relationship between the two tangents from an external point?
Both tangents have equal length, and they are symmetric about the line connecting the external point to the circle's center. The angle between them can be calculated using trigonometry.