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Tangent Circles — Definition, Formula & Examples

Tangent circles are two circles that touch each other at exactly one point. They can be externally tangent (touching on the outside) or internally tangent (one circle touching the other from the inside).

Two circles are tangent if and only if they intersect at precisely one point. If the distance between their centers equals the sum of their radii, the circles are externally tangent. If the distance between their centers equals the absolute value of the difference of their radii, the circles are internally tangent.

Key Formula

d = r_1 + r_2 \quad \text{(externally tangent)}$$ $$d = |r_1 - r_2| \quad \text{(internally tangent)}
Where:
  • dd = Distance between the centers of the two circles
  • r1r_1 = Radius of the first circle
  • r2r_2 = Radius of the second circle

How It Works

To determine whether two circles are tangent, you compare the distance between their centers to their radii. Calculate the center-to-center distance dd using the distance formula. If d=r1+r2d = r_1 + r_2, the circles are externally tangent — they sit side by side and just barely touch. If d=∣r1−r2∣d = |r_1 - r_2|, one circle sits inside the other and they share exactly one point of contact (internally tangent). At the point of tangency, the two circles share a common tangent line that is perpendicular to the line segment connecting their centers.

Worked Example

Problem: Circle A has center (1, 2) and radius 3. Circle B has center (9, 8) and radius 7. Determine whether the circles are tangent and, if so, what type.
Step 1: Find the distance between the centers using the distance formula.
d=(9−1)2+(8−2)2=64+36=100=10d = \sqrt{(9-1)^2 + (8-2)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
Step 2: Check for externally tangent: compute the sum of the radii.
r1+r2=3+7=10r_1 + r_2 = 3 + 7 = 10
Step 3: Compare: since d=r1+r2=10d = r_1 + r_2 = 10, the circles are externally tangent.
Answer: The circles are externally tangent because the distance between their centers (10) equals the sum of their radii (3 + 7 = 10).

Another Example

Problem: Circle P has center (0, 0) and radius 8. Circle Q has center (3, 4) and radius 3. Determine whether the circles are tangent.
Step 1: Find the distance between the centers.
d=32+42=9+16=5d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
Step 2: Check for internally tangent: compute the absolute difference of the radii.
∣r1−r2∣=∣8−3∣=5|r_1 - r_2| = |8 - 3| = 5
Step 3: Since d=∣r1−r2∣=5d = |r_1 - r_2| = 5, the circles are internally tangent. Circle Q sits inside Circle P, and they touch at one point.
Answer: The circles are internally tangent because the distance between centers (5) equals the absolute difference of radii (|8 − 3| = 5).

Why It Matters

Tangent circles appear frequently in high school geometry and on standardized tests like the SAT and ACT, where you must analyze how circles relate to each other in the coordinate plane. Engineers and designers use tangent circle relationships when creating gears, cam mechanisms, and rounded junctions in CAD software. Mastering this concept also builds a foundation for problems involving common tangent lines and circle packing.

Common Mistakes

Mistake: Confusing externally tangent and internally tangent by mixing up sum vs. difference of radii.
Correction: Externally tangent means d=r1+r2d = r_1 + r_2 (circles on the outside of each other). Internally tangent means d=∣r1−r2∣d = |r_1 - r_2| (one circle inside the other). Draw a quick sketch to see which case applies.
Mistake: Forgetting to take the absolute value when checking for internal tangency.
Correction: Always use ∣r1−r2∣|r_1 - r_2| so the result is positive regardless of which radius is larger.

Related Terms