quartercircles

Quarter circles are a type of circular sector obtained by dividing a circle into four congruent parts. A quarter circle is the sector with a central angle of 90 degrees (pi/2 radians), bounded by two radii that meet at a right angle and by the arc of the circle between them. In other words, it is one quarter of a full circle.

If the circle has radius r, the area of a quarter circle is A = (pi r^2)/4, and the length of its curved edge (the arc) is s = (pi r)/2. The boundary also includes the two straight radii, so the total perimeter is P = 2r + (pi r)/2.

Geometrically, a quarter circle can be formed by dividing a circle with two perpendicular radii. In the Cartesian plane, the quarter circle in the first quadrant is the region of points (x, y) satisfying x^2 + y^2 <= r^2 with x >= 0 and y >= 0. The shape is symmetric about the line y = x when centered at the origin.

Related concepts include the sector, which is any portion of a circle bounded by two radii and an arc, and the circular segment, bounded by an arc and a chord. Quarter circles frequently appear in tiling patterns, architectural rounding of corners (fillets), and geometric problems involving area and arc length.