Circle-Circle Tangents — Definition, Formula & Examples
Circle-circle tangents are straight lines that are tangent to two circles at the same time, touching each circle at exactly one point without crossing into its interior.
Given two circles with centers , and radii , , a common tangent is a line that is tangent to both circles. An external common tangent does not pass between the circles, while an internal common tangent crosses the segment joining their centers. Two non-overlapping circles of different radii have up to four common tangent lines, depending on their relative positions.
Key Formula
L_{\text{ext}} = \sqrt{d^2 - (r_1 - r_2)^2}$$
$$L_{\text{int}} = \sqrt{d^2 - (r_1 + r_2)^2}
Where:
- = Length of the external common tangent segment between the two tangent points
- = Length of the internal common tangent segment between the two tangent points
- = Distance between the centers of the two circles
- = Radii of the two circles
How It Works
To find circle-circle tangents, you need the centers and radii of both circles. External tangents touch the same side of both circles (both tangent points are on the "outside"), while internal tangents cross between the circles. The number of common tangents depends on how the circles are positioned: two separate circles have 4 tangents, externally tangent circles have 3, overlapping circles have 2, internally tangent circles have 1, and one circle inside the other has 0. You can find the length of each tangent segment using the distance between centers and the two radii.
Worked Example
Problem: Two circles have radii 5 and 3, and their centers are 12 units apart. Find the length of an external common tangent and an internal common tangent.
Step 1: Identify the given values: , , and .
Step 2: Apply the external tangent length formula.
Step 3: Check that an internal tangent exists. Since , the circles are separate and internal tangents exist.
Step 4: Apply the internal tangent length formula.
Answer: The external common tangent has length units, and the internal common tangent has length units.
Another Example
Problem: Two circles have radii 7 and 3, and their centers are 10 units apart. How many common tangent lines do they have?
Step 1: Compare the distance with and .
Step 2: Since , the circles are externally tangent — they touch at exactly one point on the outside.
Step 3: Externally tangent circles have exactly 3 common tangents: 2 external and 1 internal (passing through the point of tangency).
Answer: The two circles have exactly 3 common tangent lines.
Why It Matters
Circle-circle tangents appear in high school geometry and competition math problems involving tangent lines, radical axes, and circle packing. Engineers use common tangent calculations when designing belt-and-pulley systems, where the belt wraps around two circular pulleys and the straight belt segment is a common tangent. Understanding tangent configurations also builds intuition for conic sections and inversive geometry in college-level coursework.
Common Mistakes
Mistake: Using in the external tangent formula instead of .
Correction: For external tangents, the radii are subtracted: . For internal tangents, the radii are added: . Think of it this way — external tangents exist even when circles nearly touch, so the subtracted term must be the smaller one.
Mistake: Trying to compute internal tangent length when the circles overlap or one contains the other.
Correction: Internal tangents only exist when (circles are separate). If , the expression under the square root becomes zero or negative, indicating no internal tangent exists.