Connect the circle centers

When a circle touches a straight floor, its center is one radius above the floor. When two circles touch externally, the straight distance between their centers is the sum of their radii.

For circles of radii 4 and 1 on the same floor, the centers are 4 + 1 = 5 apart. Their heights differ by 4 − 1 = 3. The horizontal distance is 4, making a 3–4–5 right triangle.

Keep these three distances separate: the horizontal distance, the difference in heights, and the straight center-to-center distance. Diagrams in this practice are not to scale; use the stated lengths.

Now the left radius R is 9 and the horizontal distance d is 12. The unknown circle is to the right, above the same floor, and touches the given circle externally.

The center distance is 9 + r. The height difference is 9 − r. Apply Pythagoras: 12² + (9 − r)² = (9 + r)².

The square terms cancel when you expand both sides, leaving 144 = 36r. Divide by 36 to get r = 4.

Check it: the centers differ in height by 9 − 4 = 5, and their straight distance is 9 + 4 = 13. The equation 12² + 5² = 13² holds. In general, the same reasoning gives d² = 4Rr.

The given circle has radius 16. The horizontal distance between centers is 8. Both circles are above the floor, the unknown circle is to the right, and they touch externally.

Use d² = 4Rr with these new values. After finding r, check the sum of the radii against the straight center distance.

This lesson offers two choices to check the method. The practice gives five choices and later introduces corners, diagonal boundaries and a circle between two unequal circles.