A rounded rectangle is the shape obtained by taking the convex hull of four equal circles of radius 
and placing their centers at the four
corners of a rectangle with side lengths 
and 
.
A filled rounded rectangle with 
(or 
) is called a stadium.
The rounded rectangle has perimeter
 |
(1)
|
A filled rounded rectangle has area
 |
(2)
|
For a rounded square with (center) length and width 
, the corner radius can be determined by measuring the generalized
diameter 
from the edge of one rounded corner to the diagonally opposite corner. From the Pythagorean theorem, the diagonal distance 
from the rounded corner to the corner
of the circumscribed square is
 |
(3)
|
and the corner radius 
is related to the edge length of the square circumscribing the corner circle by
 |
(4)
|
Combining gives
 |  |  |
(5)
|
 |  |  |
(6)
|
