1. What is Tangent Length?

The tangent length is the distance from a point outside a circle to the point where the tangent line touches the circle. A tangent is a line that touches the circle at exactly one point.

2. How Does the Calculator Work?

The calculator uses the tangent length formula:

Where:

• \( d \) — Distance from the point to the center of the circle
• \( r \) — Radius of the circle

Explanation: This formula comes from the Pythagorean theorem applied to the right triangle formed by the radius, tangent line, and line connecting the point to the center.

3. Practical Applications

Details: Tangent calculations are used in engineering (gear design), architecture (round structures), physics (circular motion), and computer graphics (circle rendering).

4. Using the Calculator

Tips: Enter both distance and radius in the same units. The distance must be greater than the radius for a tangent to exist.

5. Frequently Asked Questions (FAQ)

Q1: What if distance equals radius?
A: When d = r, the point lies on the circle and tangent length would be zero (the point itself is the tangent point).

Q2: What if distance is less than radius?
A: The point is inside the circle and no tangent exists (result will be imaginary).

Q3: Does this work for 3D spheres?
A: The same formula applies for tangent plane distance to a sphere when considering 2D cross-sections.

Q4: Can I use different units?
A: Yes, as long as both distance and radius use the same units (e.g., both in cm or both in inches).

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, limited only by the precision of your input values.