How to sew inset circles
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I was very nervous the first time I sewed an inset circle, but the difficulty laid in the mental hurdle of doing something new, combined with my long-standing reservation towards sewing curves.
The thing that really helped me was realizing that an inset circle is essentially the exact same thing as strip-piecing four quarter circles together.
Take a look.

If you sew four quarter circles in a row, you are essentially sewing a circle, but taking little breaks along the way. For inset circles, you're sewing the circle in one go - efficiency!
To begin, we need to mark our registration points. I like marking 8 spots along my circle and outer fabric piece, with 45 degrees between each mark as shown below. Mark the spots by either folding and creasing the fabric or use a water soluble pen.

Next, match the registration marks, right sides together, and pin in place.
For the inset circle, place the outer fabric on top of the circle, right sides facing up. Now imagine the outer piece getting flipping in on itself - each corner is going to end up in the middle so that the concave edge can be matched with convex edge of the yellow circle, right sides together.


It's weird, it's a little awkward, that's learning something new.
If you need a video visual, here's a video of me pinning both the quarter circles and inset circle.
Now it's time to sew a circle, but first we need to set up our machine for success. The biggest thing that made sewing curves easier for me was when I changed out my bulky quarter inch foot for the sleek HP foot. The narrow profile of the foot helps glide around those corners without adding friction to the fabric. I also like using a coordinating thread just in case something goes wonky and the thread peeks through a bit.

I like starting to sew right before a pin, and then sew all the way around the circle. Some people recommend back stitching at the start and end, but I just sew over my starting stitches for ~0.25-0.5".

Next, press the seam either direction, but I don't recommend pressing it open.

And there you go - you have an inset circle!
Inset circle math
Want to make your own inset circles? The math is pretty basic.
For a finished inset circle of diameter n",
- diameter of inset circle: n" + 0.5"
- diameter of circle cut from outer fabric: n" - 0.5"

Let me know how it goes for you!