An oval is a curve resembling a squashed circle but, unlike the ellipse, without a precise mathematical definition. The word oval derived from the Latin word "ovus" for egg. Unlike ellipses, ovals sometimes have only a single axis of reflection symmetry (instead of two).
The particular variant illustrated above can be constructed with a compass by joining together arcs of different radii such that the centers of the arcs lie
on a line passing through the join point (Dixon 1991). Albrecht Dürer used this
method to design a Roman letter font. If the separation between left and right caps
is 
and the radii are 
and 
,
respectively, with 
and 
, then the center 
and radius 
of the joining circle are
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(1)
|
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(2)
|
Call the three circles 
,
, and 
. Let the upper point of intersection of 
and 
be 
, let the angle between the vertical
dashed line and the line through 
be 
, and let the angle between the horizontal
radius of 
and dashed line through 
be 
. Then
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(3)
|
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(4)
|
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(5)
|
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(6)
|
and half the area enclosed by the oval is the sum of the areas of leftmost quarter-circle, the sector of 
,
and the sector of 
minus the area of the triangular portion of the sector of 
lying below the 
-axis, so
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  | )/(2(R-r)^2)tan^(-1)((2a(R-r))/(a^2-(R-r)^2))].](/images/equations/Oval/Inline49.svg) |
(9)
|
As expected, this formula reduces to the area of a circle
 |
(10)
|
for 
, and to the area of a stadium
 |
(11)
|
for 
.
