How to calculate the circumference and area of an ellipse

The circumference of the ellipse: l = t (r + r), the area: s = a b (where a and b are the semi-major axis of the ellipse and the length of the semi-minor axis respectively).

Area formula: s= (pi) a b perimeter formula: l = [0, 2]4a*sqrt(1-(e*cost) 2)dt 2 (a 2+b 2) 2) [approximate perimeter of the ellipse], where a is the major semi-axis of the ellipse and e is the eccentricity (approximate algorithm).

Ellipse circumference formula.

I have seen many posts discussing the circumference of the ellipse, and I have transcribed the formula below. Sometimes you can measure it on a graph, and sometimes it's easy to calculate. If you are writing a program, you need to use the exact formula:

According to the standard elliptic equation: major semi-axis a, minor semi-axis b.

Let =(a-b) (a+b), the ellipse circumference l:

l=π（a+b）（1 + 2/4 + 4/64 + 6/256 + 25λ^8/16384 +

Simplification: l [sqrt(ab)] or.

l≈π（a+b）（64 - 3λ^4)/（64 - 16λ^2)

Explanation: 2 denotes the square of , and so on.

It is enough to take the first two items of the series.

The area of the ellipse.

As shown in Figure 3 7, o is called the center of the ellipse, a, a, b, b is called the "vertice", aa is called the "major axis", and bb is called the "minor axis".

In addition, the long OA A is called the "long radius" and the short OB is called the "short radius".

There are also those who call the ellipse "oblong".

When a b, the ellipse is a circle.

When denoting the area of an ellipse as s, the area of the ellipse can be found using the formula s ab. a b, of course, s represents the area of the circle.

When the long radius A3 (cm) and the short radius B2 (cm), its area is S3 2 6 (cm2).

In the examples so far, such as the length of the circumference, the length of the arc, the area of the circle, the area of the fan, the area of the bow, the area of the ellipse, etc., all use pi.

In this way, it is indispensable to calculate not only circles but also ellipses.

The circumference of the ellipse is 5 8 and the area is 5. The formula for ellipse circumference: l=2 b+4(a-b).

Ellipse area formula: s = a b, where a and b are the semi-major axis of the ellipse, and the length of the semi-minor axis respectively.

The ellipse is the sum of the distances from the plane to the fixed points f1 and f2 equal to the constant (greater than |f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is: |pf1|+|pf2|=2a（2a>|f1f2|）。

An ellipse is a closed conical section: a planar curve that is intersected by a cone and a plane. The ellipse has many similarities with the other two forms of conical sections:

Parabola and hyperbola, both are open and unbounded. The cross-section of a cylinder is elliptical unless the section is parallel to the axis of the cylinder.

In fact, the area formula of the ellipse is pie multiplied by a times b, this a is the semi-major axis, b is the semi-minor axis, and the perimeter formula can be calculated by line integral.

The formula for ellipse circumference: l=2 b+4(a-b).

Ellipse area formula: a = (pi) a b, where a and b are the semi-major axis of the ellipse, and the length of the semi-minor axis, respectively.

For this question, a = cm, b = 1 cm

Substitute the perimeter and area formulas above.

l=2π+4× cm

a=π× cm^2

That is, the circumference of the ellipse is 2 + 6 cm, and the area is square centimeters.

The formula for ellipse circumference is l=2 b+4(a-b). The elliptic circumference theorem is that the circumference of an ellipse is equal to the circumference (2b) of the ellipse with the length of the minor semi-axis of the ellipse as the radius, plus four times the difference between the length of the major semi-axis and the length of the minor semi-axis of the ellipse. Formula Description:

In the formula, a represents the length of the major semi-axis of the ellipse, b represents the length of the minor semi-axis of the ellipse, which is the pi, and l represents the circumference of the ellipse.

Ellipse Area Formula:

s= (pi) a b, where a and b are the major semi-axis of the ellipse and the length of the minor semi-axis, respectively.

Ellipse formula: (x-h) a + (y-k) b = 1. Formula description: In the formula, a and b are the length of the long and short axes, the center point is (h, k), and the main axis is parallel to the x-axis.

The standard equation for an ellipse.

The standard equation for an ellipse is divided into two cases:

When the focus is on the x-axis, the standard equation for an ellipse is: x 2 a 2+y 2 b 2=1,(a>b>0);

When the focus is on the y-axis, the standard equation for an ellipse is: y2 a 2+x 2 b 2=1,(a>b>0);

where a2-c 2=b 2;

Derivation: PF1+PF2>F1F2 (p is the point on the ellipse, f is the focal point).

Nature of Ellipse:

1. Symmetry: symmetry on the x-axis, symmetry on the y-axis, symmetry on the center of the origin.

2. Vertices: (a,0)(-a,0)(0,b)(0,-b).

3. Eccentricity: e= (1-b 2 a).

4. Eccentricity range: 05. The smaller the eccentricity, the closer it is to the circle, and the larger the ellipse, the flatter the ellipse.

6. Focus (when the center is the origin) :(c,0),(c,0) or (0,c),(0,-c).

7. p is a point on the ellipse, a-c pf1 (or pf2) a+c.

8. The circumference of an ellipse is equal to the length of a particular sinusoidal curve in a period.

Focal radius. Focus on the x-axis: |pf1|=a+ex |pf2|=a-ex(f1, f2 are the left and right focus, respectively).

The radius of the ellipse over the right focal point r=a-ex.

The radius of the left focal point r=a+ex.

Focus on the y-axis: |pf1|=a+ey |pf2|=a-ey(f2,f1 are the upper and lower focus, respectively).

The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) of the focal point and the two intersections of the ellipse a,b, i.e., |ab|=2*b^2/a。

The formula for ellipse circumference: l=2 b+4(a-b).

According to the first definition of an ellipse, a is used to denote the length of the major semi-axis of the ellipse, b is the length of the minor semi-axis of the ellipse, and a>b>0.

Ellipse circumference theorem: The circumference of an ellipse is equal to the circumference of the ellipse whose minor semi-axis length is the radius (2 b) plus four times the difference between the length of the major semi-axis of the ellipse (a) and the length of the minor semi-axis (b).

Ellipse Area Formula: s= ab

Ellipse area theorem: The area of an ellipse is equal to the product of pi ( ) multiplied by the length of the major semi-axis of the ellipse (a) and the length of the minor semi-axis (b).

Area formula for an ellipse.

s= (pi) a b, where a and b are the major semi-axis of the ellipse and the length of the minor semi-axis, respectively.

or s= (circumferential rate) capacity a b 4 (where a and b are the major axis of the ellipse and the length of the minor axis respectively).

The formula for the circumference of an ellipse.

There is no formula for elliptic circumference, there are integral or infinite nomials.

The exact calculation of the circumference of an ellipse (l) requires the summation of integrals or infinite series. As.

l = [0, 2]4a * sqrt(1-(e*cost) 2)dt 2 (a 2+b 2) 2) [approximate circumference of the ellipse], where a is the major semiaxis of the ellipse and e is the eccentricity.

The eccentricity of the ellipse is defined as the ratio of the distance from the point on the ellipse to a focal point and the distance from the point to the corresponding alignment of the focal point.

e=pf/pl

Some of the common secondary conclusions in the ellipse are as follows:

1. The elliptical eccentricity is defined as the ratio of the focal length on the ellipse to the major axis, (range: 02c. The greater the eccentricity, the flatter the ellipse; The smaller the eccentricity, the closer the ellipse is to the circle.

2. The focal distance of the ellipse: the distance between the focal point of the ellipse and its corresponding alignment (such as the focal point (c,0) and the alignment x= a 2 c) is a 2 c-c=b 2 c.

3. Focus on the x-axis: |pf1|=a+ex |pf2|=a-ex(f1, f2 are the left and right focus, respectively).

4. The radius of the ellipse over the right focal point r=a-ex.

5. The radius of the left focal point r=a+ex.

The focal triangle properties of an ellipse are:

1）|pf1|+|pf2|=2a。

2）4c²=|pf1|²+pf2|²-2|pf1|·|pf2|·cosθ。

3) Circumference = 2a + 2c.

4) Area = S=B·tan(2)(F1PF2=).