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?
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$\begingroup$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).
$\endgroup$ 8 $\begingroup$Each of the four shapes is called a circular segment.
To find the perimeter of a segment, you need to add the length of its arc to the length of the chord. Both of them can be calculated from the length of the radius and the length of the central angle $\theta$.
Similarly, you find the area by finding the area of the circular sector (which is the segment plus the triangle), which is easily calculated from the area of the circle and the central angle. Then you subtract the area of the triangle, which can be found from the radius and the central angle using trigonometry.
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