Find the Tangent or How to Sew a Circle into a Straight Line June 13 2015

Sewing a pillbox hat, where a round cap is stitched to a straight edge, is an excercise in finding the tangent.

Math teachers describe the tangent to a circle by drawing  a slash on the curved edge. Really though, it's 'a' tangent, not 'the' tangent. 

There are tangents all around the circle, and in sewing, each is as long as a single stitch. 

The circumference of the circle to be sewed--assuming a 5/8" seam allowance--is cut to exactly the same length of the straight piece whose ends are joined to make an open cylinder. Both equal 2 x π x the radius of the circle. Pi, π, is about equal to 3.14; the diameter is 2 x the radius. So for a size 22" hat, the diameter of the top is 7" and its circumference is the same length as the straight piece. D=22"/3.14, or 7".

The two fabric pieces being joined need lay flat only where the stitch is made; don't be concerned if the fabric pieces folds elsewhere, which it tends to do. Fitting a circular piece to a straight piece is less daunting when considering that all one need do is find that single-stitch tangent. 

If the circular top piece puckers more than the straight piece then the circumference is longer than the straight piece length. To compensate, add 1/8th to the circular tops seam allowance while keeping the bottom straight piece at 5/8". 

Pi every day

When sewing any kind of curve, use Pi to make adjustments. For a more complicated explanation of finding a tangent, see