How can I make an image like this?
When I try doing this in Adobe Illustrator, it ends up like this. the final few circles overlap with the first few. How do I avoid this?
Let me suggest a solution with a slightly better workflow. Recognize that the same shape repeats several times so let us first make that one shape. So:
Rotate circle by 360/8° from a corner point (Alt+LMB click). (replace 8 with any amount of petals you want)
Image 1: Rotate one copy
Use shape builder or pathfinder to carve out the one shape.
Image 2: Make one petal with shapebulilder. (or use pathfinder)
Rotate object by 360/8° from corner, then hit ctrl+D repeatedly until you have a full circle.
Image 3: Replicate by repeating last transform
Alright alright, do it like this:
Step 2: Arrange the first circle on top of the others (Right click > Arrange > Bring to Front).
Step 3: Select the last circle and the first one, click on the shape builder.
Step 4: Click on the last circle, this will cut it.
Step 5: Arrange the first circle to the bottom of the others (Right click > Arrange > Send to Back.
Step 6: You see the last circle will have a cut, it's 2 pieces now.
Delete the small piece and there you have it.