If you inscribe a square inside a circle what is the name of the shape in the circle but outside of the square When you inscribe a square in a circle there is part of the circle that is not in the square. There are $4$ of these shapes. What is the name of these shapes and what is their area and perimeter?

Each of the four congruent shapes is called a minor segment of the circle. A segment is demarcated by a chord of a circle and the arc subtended by the chord. The chord divides the circle into the minor (smaller) and major (larger) segments, except when the chord is a diameter, in which case it divides the circle into two equal semicircles. So the semicircle is the special case of a segment.

In this case, each segment is subtended by a central right angle, and its area is given by $A = \frac 12 r^2(\theta - \sin\theta) =\frac 12 r^2(\frac{\pi}{2} - 1)$.

(note that the angle measure is in radians).

The perimeter of each segment is $r(\frac{\pi}{2}+ \sqrt 2) $ (the former is the arc length term, the latter is the side of the inscribed square by the Pythagorean theorem).