The ellipse area.

The elliptical segment area.

Perimeter of the ellipse

(notes by R. Bigoni)

From the canonical equation of the ellipse we can easily deduce the expressions of
the cartesian coordinates of a point *P* of the ellipse as functions of the angle α
formed by the vector *OP* (where *O* is the origin of the reference frame)
and the abscissa axis.

In fact, if we let and square both the sides of both equations, by summing the squared equations we obtain the canonical equation

So we have

which are the **parametric equations of the ellipse**.

The parametric equations of *x* e *y* can be respectively interpreted
as the abscissa and the ordinate of the points *A* and *B* intercepted by the ray,
starting at *O* and forming the angle α with respect the x-axis, on the
concentrical circumferences with radii *a* and *b*.

The point *E* in the figure, which has the same abscissa as *A* and the
same ordinate as *B*, is a point of the ellipse with semiaxes *a* and *b*.

The ordinates *y*_{E} e *y*_{A} of the points *E* and
*A* are

so we have

that is the ratio between the ordinate of the point *E* of the ellipse with semiaxes
*a* and *b* and the ordinate of the point *A* with same abscissa of the
circumference with same center and radius *a* is
^{b}/_{a}.

If, considering for simplicity sake only the points with positive ordinate, for each
point *E* and *A* we construct the rectangles on the same base *dx* and
altitudes *y*_{E} e *y*_{A}, the former rectangles have
area ^{b}/_{a} with respect to that of the latter ones.

Consequently, the sum of all the first rectangles is
^{b}/_{a} with respect to the sum of the second ones.
If *dx* is infinitesimal, the first sum is the area of half the ellipse,
the second one is the area of half the circle. In conclusion, the area of half the ellipse
is ^{b}/_{a} with respect to the area of half the
circle and, obviously, the area of the ellipse is ^{b}/_{a}
with respect to that of the circle.

A straight line parallel to the y-axis with equation
delimitates the figure *EE'V* (colored in blue), which is said **elliptical right
segment**. The area ε of this figure is
^{b}/_{a} with respect to the area η of the circular segment
*AA'V*.

The area η in its turn can be obtained by subtracting from the area σ of the circular sector
*AOA'V* the area τ of the triangle *AOA'*

The area σ is to circle area as its central angle *AOA'* is to full angle,
that is as its half *α=AOV* is to the straight angle.

The area τ measures

therefore the area η measures

and the area ε measures

In order to express this area as function of *h*, we notice that

therefore

Example.

Given the ellipse γ with semiaxes *a=4* and *b=3*, the areas of the
elliptical segments, obtained by cutting γ parallelly to minor axis at the distance
*h=2* from it, have measures

We can demonstrate that the perimeter of the ellipse with semimajor axis *a* and eccentricity *e* is

The following Javascript application allows you to approximate the perimeter of the ellipse.