Ellipse Perimeter

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What is an Ellipse?

An ellipse can be described as a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

An ellipse has two types of axis – Major Axis and Minor Axis. The longest chord of the ellipse is known as the major axis. The perpendicular chord to the major axis is known as the minor axis which bisects the major axis at the centre.


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Ellipse Formula

As we know, we can define an ellipse as a closed-shape structure in a two-dimensional plane. Hence, the ellipse covers a region in a 2D plane. So, this bounded region of the ellipse is the area of the ellipse.  The shape of the ellipse is different from that of the circle, hence the formula for its area will also be different.

Let’s discuss about the area and the perimeter of ellipse.

Area of Ellipse

The area of the circle is calculated based on its radius, but the area of ellipse depends on the length of the minor axis and major axis.

Area of the circle = πr2

And,

Area of the ellipse = Pie(π) x Semi-Major Axis x Semi-Minor Axis


Area of the ellipse = π.a.b

Where the value of pie (π) = 22/7 or 3.14


Perimeter of Ellipse

The perimeter of an ellipse can be defined as the total distance run by its outer boundary. For a circle, it is very easy to find its circumference, since the distance from the centre to any point of locus of a circle is the same. This distance is called the radius.

But in the case of an ellipse, we have two axis, the major and minor axis, that crosses through the centre and intersects. Hence, an approximation formula can be used to find the perimeter of an ellipse :

The perimeter of Ellipse = \[2 \pi \sqrt{\frac{a^{2} + b^{2}}{2}}\]

Where a is the length of the semi-major axis and b is the length of the semi-minor axis respectively.


What is Latus Rectum?

The line segments that are perpendicular to the major axis through any of the foci such that their endpoints lie on the ellipse are defined as the latus rectum.


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The length of the latus rectum is 2b2/a.

L = 2b2/a

Where a is the length of the minor axis and where b is the length of the major axis.


Solved Examples

Question 1) If the length of the semi-major axis is  8 cm and the semi-minor axis is 5cm of an ellipse. Find its area.

Solution)  Given, length of the semi-major axis of an ellipse, a = 8cm

length of the semi-minor axis of an ellipse, then b equals 5cm

By the formula of area of an ellipse, we know that;

Area = π × a × b

Area = π × 8 × 5

Area = 40 π

or

Area = 40 × 22/7

Area = 110 cm2


Question 2) If the length of the semi-major axis is 10 cm and the semi-minor axis is 5 cm of an ellipse. Find the perimeter of ellipse.

Answer) Given, length of the semi-major axis of an ellipse, a = 10 cm

length of the semi-minor axis of an ellipse, b equals 5cm

By the formula of Perimeter of an ellipse, we know that;

The perimeter of ellipse = \[2 \pi \sqrt{\frac{a^{2} + b^{2}}{2}}\]

Therefore, the Perimeter of ellipse = \[2 \times 3.14 \sqrt{\frac{10^{2} + 5^{2}}{2}} = 49.64\]


Fun Facts

  1. The ellipse was first studied by Menaechmus, investigated by Euclid, and was named by Apollonius. The focus and conic section directrix of an ellipse was considered by Pappus. In 1602, it was Kepler who believed that the orbit of Mars was oval; he later discovered that it was an ellipse with the Sun at one focus.

  2. An ellipse and a circle are both known to be examples of conic sections.

  3. A circle is known as a special case of an ellipse, with the same radius for all points.

  4. By stretching a circle in the x-direction or y-direction, an ellipse is created.

  5. An ellipse is formed when a plane intersects a cone at an angle to its base.

  6. All ellipses have two focal points or foci. The sum of the distances from every point on the ellipse to the two foci is known to be constant.

  7. All ellipses have a centre, a major axis, and a minor axis.

  8. All ellipses have eccentricity values greater than zero or equal to zero, and less than one.

  9. Real-Life Examples of Ellipse Many real-world situations can be represented by ellipses, including orbits of various planets, satellites, moons, and comets, and shapes of boat keels, rudders, and some aeroplane wings.

FAQ (Frequently Asked Questions)

Question 1. How do you Find the Perimeter of an Ellipse?

Answer:

  1. When a equals b, the ellipse is a circle, and the perimeter of an ellipse is 2πa (62.832... in our example).

  2. When b equals 0 (the shape is two lines back and forth) the perimeter of an ellipse is 4a (40 in our example).

Question 2. How do you Find the Perimeter?

Answer: To find the perimeter of a rectangle, add the lengths of the rectangle's four sides. If you have only the width and the height, then it is very easy to find all four sides (two sides are each equal to the height and the other two sides are equal to the width). You need to multiply both the height and width by two and add the results.

Question 3. What is the General Equation of Ellipse?

Answer: The standard equation for an ellipse, x2/a2 + y2/b2 = 1, represents an ellipse centred at the origin and with axes lying along the coordinate axes. In general, an ellipse may be centred at any point, or have axes that are not parallel to the coordinate axes.

Question 4. How you Find the Perimeter of a Rectangle?

Answer: The perimeter of a rectangle can be calculated by the total length of all the sides of the rectangle. Hence, we can find the perimeter by adding all four sides of a rectangle. The perimeter of the given rectangle is a + b + a + b.

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