Area of Ellipse

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Introduction to Area of Ellipse

An ellipse just seems to be like an ordinary oval shape, resulting when a cone is snipped off by an oblique plane in a way that yields a closed curve which does not bisect the base. The area of ellipse is the same as the area of a circle i.e. A = π*r² or also written as: A= π · r * r. If a circle becomes flat it transforms into the shape of an ellipse and the semi-axes (OA and OB) of such an ellipse will be the stretched and compressed radii.


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Formulas for Ellipse

In an ellipse, if ‘a’ stands for OA and ‘b’ stands for OB, it can be proved that the area of the ellipse can be calculated by substituting ‘ab’ for ‘r r’ in the formula for the area of the circle, which then provides the below given:

Formula for the area of an ellipse:  A = π · a · b

Where ‘a’ (horizontal segment) = major axis [semi-major axis or ½ of the major axis]

    ‘b’ (vertical segment)   = minor axis [semi-minor axis or ½ the minor axis]

\[\text{Formula for Perimeter of the ellipse is: P = } 2 \pi \sqrt{\frac{a^{2} + b^{2}}{2}}\]

\[\text{Formula for volume of the ellipse is: V = } \frac{\pi (R_{1} + R_{2} + R_{3})}{3}\]


Major and Minor Axes

The Major Axis is said to be the longest diameter of an ellipse. This line segment travels from one side of the ellipse, through the center, to the other side, at the broadest part of the ellipse. And the Minor Axis is known to be the shortest diameter (at the narrowest part of the ellipse).

The half of the Major Axis is called the Semi-major Axis, and half of the Minor Axis is the Semi-minor Axis.

Below is a clear depiction of major and minor Axis with formulas to calculate:

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Special Case of Ellipse in a Circle's Area

A circle is a unique case of an ellipse. In correspondence to the fact that a square is a kind of rectangle, a circle is also a special case of an ellipse. The formula used to calculate the area of a circle is π r². However, seeing that a circle is an ellipse with equal minor and major axes, the formula for the area of the ellipse is the same as the formula for area of a circle.


Purpose of Calculating Area of Ellipse

Area of ellipse can be used to calculate a number of figures and fields such as:-

  • Find the surface area of a pond or an oval pool cover.

  • Computing the volume of a large table or an oval table top.

  • Compute the surface area of an oval structure like a dome.

  • Computing the area of a Football field.

  • Constructing an agricultural tunnel or an Elliptical Pipe.

  • To set a comparison between round and oval furnace hole airflow.

  • Used in laying flooring.

Solved Examples

Example 1:

A billiards table has horizontal radius of measurement 16.5 cm and vertical radius 10.5 cm. Calculate the area of the ellipse.


Solution1:

A billiards table is in the shape of an ellipse.

Given that a billiards table has:

Horizontal radius (a) = 16.5 cm

Vertical radius (b) = 10.5 cm

Area of ellipse = π × a × b

Thus,

A = π × 16.5 × 10.5

= 22/7 × 173.25

=544.5

Therefore,

A = 544.5cm2

Note: we will take the value of π as 22/7 unless, otherwise stated.


Example 2:

Determine the volume, area and perimeter of an ellipse having the radius R1, R2 & R3 of 11 cm, 8 cm, and 3 cm respectively?


Solution2:

Following are the given values:-

radius R1 = 11 cm

radius R2 = 8 cm

radius R3 = 3 cm

Step by step calculation

Using the Formula to calculate the volume = (4π/3) x R1 x R2 x R3

Now substituting the values

= (4π/3) x 11 x 8 x 3

= 1105.28cm3

Using the Formula to calculate area = π R1 R2

Now substituting the values

= π x 11 x 8

= 276.57 cm2

Using the Formula to calculate the perimeter = \[2 \pi (\sqrt{\frac{(R_{1} ^{2} + R_{2} ^{2})}{2}})\]

Now substituting the values

= 2π x \[\sqrt{\frac{(11 + 8)}{2}}\]

= 59.66cm2.

FAQ (Frequently Asked Questions)

1. What is an Ellipse?

Ans. The ellipse is a closed curve and is in harmony about the midpoint. In an ellipse, the distance of two points inside of an ellipse from a point on the ellipse is exactly similar to the distance of any other point on the ellipse from that same point. These points in the interior of the ellipse are known as the foci.

2. How is the Ellipse Related to Planet Pluto?

Ans. Ellipses are, by their natural characteristic, not "exactly round" which circles are technically. The measure by which an ellipse is "flattened" from being perfectly round is what we call ellipse's "eccentricity". When astronomers refer to something (like Pluto) as having an "eccentric" orbit, they don't mean that the orbit is "uncanny"; but they mean it's "far from being circular". In the case of Pluto, its orbit actually intersects inside that of Neptune from time to time and the bigger the value of Pluto’s eccentricity, the more "flattened" the ellipse.


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3. Where do we Find Ellipses in Real Life?

Ans. A physical characteristic of ellipses is that light or sound waves emerging from one focus will reflect back to the other focus. This property of ellipses is frequently put to use in medicine. For instance, a patient suffering from bladder stones can be placed next to a non-invasive clinical appliance that emits shock waves into an elliptical vessel away from the patient. This is a harmful technology where the patient can even return home the same day, having needed no invasive surgery. Also, have you ever noticed the car logo of Toyota brand or Hyundai? These are  a perfect ellipse.


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