Calculate the area of the oval along the perimeter. Oval

We invite you to try the most versatile

best

on the Internet. Our

ellipse perimeter calculator online

will not only help you find ellipse perimeter

in several ways

depending on the known data, but will also show

detailed solution

. Therefore this

ellipse perimeter calculator online

It is convenient to use not only for quick calculations, but also for checking your calculations.

Ellipse perimeter calculator online

, presented on our website, is a subsection

online calculator for the perimeter of geometric shapes

. This is why you can not only

set the calculation accuracy

, but also, thanks

easy navigation

our

online calculator

, without extra effort, proceed to the calculation perimeter
any of the following geometric shapes: triangle, rectangle, square, parallelogram, rhombus, trapezoid, circle, sector of a circle, regular polygon.

You can also literally go to

online calculator for the area of geometric shapes

and calculate square

triangle

rectangle

square

parallelogram

rhombus

trapezoids

circle

ellipse

sectors of the circle

regular polygon

also in several ways

and with

detailed solution

Ellipse

is a closed curve on a plane that can be obtained as the intersection of a plane and a circular

cylinder

, or as an orthogonal projection

circle

to the plane.

Circle

is a special case

ellipse

. Along with

hyperbole

And

parabola

ellipse

conical section

And

quadric

ellipse

is intersected by two parallel lines, then the segment connecting the midpoints of the segments formed at the intersection of the lines and

ellipse

, will always pass through

center of the ellipse

. This property makes it possible, by constructing using a compass and ruler, to obtain

ellipse center

Evoluta

ellipse

There is

asteroid

, which is stretched along the short axis.

Using this

You can do calculation of ellipse perimeter
in the following ways:

calculation of the perimeter of an ellipse through two semi-axes

;

calculation of the perimeter of an ellipse through two axes

Also using

online ellipse perimeter calculator

You can display all options presented on the site

calculating the perimeter of an ellipse

ellipse perimeter calculator online

or not, still leave comments and suggestions. We are ready to analyze every comment about the work

online ellipse perimeter calculator

and make it better. We will be glad to see every positive comment and gratitude, since this is nothing more than confirmation that our work and our efforts are justified, and

Circumference is a closed plane curve, all points of which are equidistant from a given point (the center of the circle). The distance from any point of the circle \(P\left((x,y) \right)\) to its center is called radius. The center of the circle and the circle itself lie in the same plane. Equation of a circle of radius \(R\) with center at the origin ( canonical equation of a circle ) has the form
\((x^2) + (y^2) = (R^2)\).

Equation of a circle radius \(R\) with center at an arbitrary point \(A\left((a,b) \right)\) is written as
\((\left((x - a) \right)^2) + (\left((y - b) \right)^2) = (R^2)\).

Equation of a circle passing through three points , written in the form: \(\left| (\begin(array)(*(20)(c)) ((x^2) + (y^2)) & x & y & 1\\ (x_1^2 + y_1^2) & ((x_1)) & ((y_1)) & 1\\ (x_2^2 + y_2^2) & ((x_2)) & ((y_2)) & 1\\ (x_3^2 + y_3^2) & ((x_3)) & ((y_3)) & 1 \end(array)) \right| = 0.\\\)
Here \(A\left(((x_1),(y_1)) \right)\), \(B\left(((x_2),(y_2)) \right)\), \(C\left(( (x_3),(y_3)) \right)\) are three points lying on the circle.

Equation of a circle in parametric form
\(\left\( \begin(aligned) x &= R \cos t \\ y &= R\sin t \end(aligned) \right., \;\;0 \le t \le 2\pi\ ),
where \(x\), \(y\) are the coordinates of the points of the circle, \(R\) is the radius of the circle, \(t\) is the parameter.

General equation of a circle
\(A(x^2) + A(y^2) + Dx + Ey + F = 0\)
subject to \(A \ne 0\), \(D^2 + E^2 > 4AF\).
The center of the circle is located at the point with coordinates \(\left((a,b) \right)\), where
\(a = - \large\frac(D)((2A))\normalsize,\;\;b = - \large\frac(E)((2A))\normalsize.\)
The radius of the circle is
\(R = \sqrt (\large\frac(((D^2) + (E^2) - 4AF))((2\left| A \right|))\normalsize) \)

Ellipse is a plane curve for each point of which the sum of the distances to two given points ( ellipse foci ) is constant. The distance between the foci is called focal length and is denoted by \(2c\). The middle of the segment connecting the foci is called the center of the ellipse . An ellipse has two axes of symmetry: the first or focal axis, passing through the foci, and the second axis perpendicular to it. The points of intersection of these axes with the ellipse are called peaks. The segment connecting the center of the ellipse with the vertex is called semi-axis of the ellipse . The semi-major axis is denoted by \(a\), the semi-minor axis by \(b\). An ellipse whose center is at the origin and whose semi-axes lie on coordinate lines is described by the following canonical equation :
\(\large\frac(((x^2)))(((a^2)))\normalsize + \large\frac(((y^2)))(((b^2)))\ normalsize = 1.\)

The sum of the distances from any point of the ellipse to its foci constant:
\((r_1) + (r_2) = 2a\),
where \((r_1)\), \((r_2)\) are the distances from an arbitrary point \(P\left((x,y) \right)\) to the foci \((F_1)\) and \(( F_2)\), \(a\) is the semimajor axis of the ellipse.

The relationship between the semi-axes of the ellipse and the focal length
\((a^2) = (b^2) + (c^2)\),
where \(a\) is the semi-major axis of the ellipse, \(b\) is the semi-minor axis, \(c\) is half the focal length.

Ellipse eccentricity
\(e = \large\frac(c)(a)\normalsize

Equations of ellipse directrixes
The directrix of an ellipse is a straight line perpendicular to its focal axis and intersecting it at a distance \(\large\frac(a)(e)\normalsize\) from the center. The ellipse has two directrixes located on opposite sides of the center. The directrix equations are written in the form
\(x = \pm \large\frac(a)(e)\normalsize = \pm \large\frac(((a^2)))(c)\normalsize.\)

Equation of an ellipse in parametric form
\(\left\( \begin(aligned) x &= a\cos t \\ y &= b\sin t \end(aligned) \right., \;\;0 \le t \le 2\pi\ ),
where \(a\), \(b\) are the semi-axes of the ellipse, \(t\) is the parameter.

General equation of ellipse
\(A(x^2) + Bxy + C(y^2) + Dx + Ey + F = 0\),
where \((B^2) - 4AC

General equation of an ellipse whose semi-axes are parallel to the coordinate axes
\(A(x^2) + C(y^2) + Dx + Ey + F = 0\),
where \(AC > 0\).

Ellipse perimeter
\(L = 4aE\left(e \right)\),
where \(a\) is the semimajor axis of the ellipse, \(e\) is the eccentricity, \(E\) is complete elliptic integral of the second kind.

Approximate formulas for the perimeter of an ellipse
\(L \approx \pi \left[ (\large\frac(3)(2)\normalsize\left((a + b) \right) - \sqrt (ab) ) \right],\;\;L \approx \pi \sqrt (2\left(((a^2) + (b^2)) \right)),\)
where \(a\), \(b\) are the semi-axes of the ellipse.

Area of the ellipse
\(S = \pi ab\)

In astronomy, when considering the movement of cosmic bodies in orbits, the concept of “ellipse” is often used, since their trajectories are characterized by precisely this curve. In the article we will consider the question of what the marked figure represents, and also give the formula for the length of the ellipse.

What is an ellipse?

According to the mathematical definition, an ellipse is a closed curve for which the sum of the distances from any of its points to two other specific points lying on the main axis, called foci, is a constant value. Below is a figure that explains this definition.

Formula for the circumference of an ellipse

Although the figure in question is quite simple, the length of its circumference can be accurately determined by calculating the so-called elliptic integrals of the second kind. However, the self-taught Indian mathematician Ramanujan, at the beginning of the 20th century, proposed a fairly simple formula for the length of an ellipse, which approaches the result of the marked integrals from below. That is, the value of the value in question calculated from it will be slightly less than the actual length. This formula looks like: P ≈ pi *, where pi = 3.14 is the number pi.

For example, let the lengths of the two semi-axes of the ellipse be equal to a = 10 cm and b = 8 cm, then its length P = 56.7 cm.

Everyone can check that if a = b = R, that is, an ordinary circle is considered, then Ramanujan’s formula reduces to the form P = 2 * pi * R.

Note that in school textbooks another formula is often given: P = pi * (a + b). It is simpler, but also less accurate. So, if we apply it to the case considered, we obtain the value P = 56.5 cm.

Oval is a closed box curve that has two axes of symmetry and consists of two support circles of the same diameter, internally conjugate by arcs (Fig. 13.45). An oval is characterized by three parameters: length, width and radius of the oval. Sometimes only the length and width of the oval are specified, without defining its radii, then the problem of constructing an oval has a large variety of solutions (see Fig. 13.45, a... d).

Methods for constructing ovals based on two identical reference circles that touch (Fig. 13.46, a), intersect (Fig. 13.46, b) or do not intersect (Fig. 13.46, c) are also used. In this case, two parameters are actually specified: the length of the oval and one of its radii. This problem has many solutions. It's obvious that R > OA has no upper bound. In particular R = O 1 O 2(see Fig. 13.46.a, and Fig. 13.46.c), and the centers O 3 And O 4 are determined as the points of intersection of the base circles (see Fig. 13.46, b). According to the general point theory, mates are determined on a straight line connecting the centers of arcs of osculating circles.

Constructing an oval with touching support circles(the problem has many solutions) ( rice. 3.44). From the centers of the reference circles ABOUT And 0 1 with a radius equal, for example, to the distance between their centers, draw arcs of circles until they intersect at points ABOUT 2 and O 3.

Figure 3.44

If from points ABOUT 2 and O 3 draw straight lines through the centers ABOUT And O 1, then at the intersection with the support circles we obtain the connecting points WITH, C 1, D And D 1. From points ABOUT 2 and O 3 as from centers of radius R 2 draw arcs of conjugation.

Constructing an oval with intersecting reference circles(the problem also has many solutions) (Fig. 3.45). From the intersection points of the reference circles C 2 And O 3 draw straight lines, for example, through centers ABOUT And O 1 until they intersect with the reference circles at the junction points C, C 1 D And D 1, and radii R2, equal to the diameter of the reference circle - the conjugation arc.

Figure 3.45 Figure 3.46

Constructing an oval along two specified axes AB and CD(Fig. 3.46). Below is one of many possible solutions. A segment is plotted on the vertical axis OE, equal to half the major axis AB. From the point WITH how to draw an arc with a radius from the center SE to the intersection with the line segment AC at the point E 1. Towards the middle of the segment AE 1 restore the perpendicular and mark the points of its intersection with the axes of the oval O 1 And 0 2 . Build points O 3 And 0 4 , symmetrical to the points O 1 And 0 2 relative to the axes CD And AB. Points O 1 And 0 3 will be the centers of reference circles of radius R1, equal to the segment About 1 A, and the points O2 And 0 4 - centers of conjugation arcs of radius R2, equal to the segment O 2 C. Straight lines connecting centers O 1 And 0 3 With O2 And 0 4 At the intersection with the oval, the connecting points will be determined.

In AutoCAD, an oval is constructed using two reference circles of the same radius, which:

1. have a point of contact;

2. intersect;

3. do not intersect.

Let's consider the first case. A segment OO 1 =2R is constructed, parallel to the X axis; at its ends (points O and O 1) the centers of two supporting circles of radius R and the centers of two auxiliary circles of radius R 1 =2R are placed. From the intersection points of the auxiliary circles O 2 and O 3, arcs CD and C 1 D 1 are built, respectively. The auxiliary circles are removed, then the inner parts of the support circles are cut off relative to the arcs CD and C 1 D 1. In Figure ъъ the resulting oval is highlighted with a thick line.