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# Area of Ellipse- Formula, Definition, Solved Examples

An ellipse is a flattened circle or oval shape. It is created when a cone is cut by a plane. It has a symmetrical shape with two axes:

• the major axis
• the minor axis The center of the ellipse is the midpoint of both axes. The distance from the center to the ends of each axis is called the semi-major or semi-minor axis.

### Real Life Example of Ellipse

• Planetary orbits
• Moon’s orbit around Earth
• Elliptical mirrors
• Elliptical lenses

## What Is the Area of Ellipse?

The area of an ellipse is the amount of space. It takes up within its boundary. It has a symmetrical shape with two axes, the major axis being the longest diameter and the minor axis being the shortest. The center of the ellipse is the midpoint of both axes. the distance from the center to the ends of each axis is called the semi-major or semi-minor axis.

## Area of Ellipse Formula

The formula to calculate the

area of an ellipse is A = πab,

where “a” and “b” are the lengths of the major and minor axes of the ellipse.

Note: The area of an ellipse is always positive and non-zero.

### Standard Form of an Ellipse

The standard form of an ellipse with center at the origin (0,0) is:

(x²/a²) + (y²/b²) = 1

Where:

• a is the distance from the center to the end of the major axis
• b is the distance from the center to the end of the minor axis

If the center of the ellipse is not at the origin, but instead at (h,k), the standard form becomes:

((x-h)²/a²) + ((y-k)²/b²) = 1

Where:

• (h,k) is the center of the ellipse.

## Proof of Formula of Area of Ellipse

The formula for the area of an ellipse is:

A = πab

where a and b are the lengths of the semi-major and semi-minor axes, respectively.

To prove this formula, we can start with the standard form of an ellipse:

((x-h)²/a²) + ((y-k)²/b²) = 1

where (h,k) is the center of the ellipse.

We can rewrite this equation as:

(x-h)²/a² = 1 – ((y-k)²/b²)

Then, we can solve for y:

(y-k)²/b² = 1 – (x-h)²/a²

y-k = ±(b/a)√(a² – (x-h)²)

y = ±(b/a)√(a² – (x-h)²) + k

This gives us the equation for the top and bottom halves of the ellipse in terms of x.

To find the area of the ellipse, we can integrate the equation for the top half of the ellipse from x = -a to x = a, and multiply the result by 2 to get the total area of the ellipse:

A = 2∫[k+(b/a)√(a²-(x-h)²)]dx from x=-a to x=a

We can simplify this integral by making the substitution u = (x-h)/a, which gives us:

A = 2a∫[k+(b/a)√(1-u²)]du from u=-1 to u=1

This integral can be solved using trigonometric substitution, by letting u = sinθ and du = cosθdθ, which gives us:

A = 2a[b(π/2) + a(k/2)∫cos²θdθ] = πab

Therefore, the formula for the area of an ellipse is proven to be A = πab.

## How to Find the Area of Ellipse?

To find the area of an ellipse, follow these steps:

1. Measure the length of the major axis of the ellipse.
2. Measure the length of the minor axis of the ellipse.
3. Divide both the length of the major and minor axis by 2 to get the semi-major and semi-minor axes.
4. Square the semi-major axis and semi-minor axis and multiply them by π.
5. The resulting product is the area of the ellipse.

Here is the formula in mathematical notation:

A = πab

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

## Solved Examples on Area of Ellipse

Example 1: Find the area of an ellipse with semi-major axis length of 5 and semi-minor axis length of 3.

Solution: We can use the formula for the area of an ellipse, A = πab,

where a = 5 and b = 3.

we get:

A = π(5)(3) = 15π

Therefore, the area of the ellipse is 15π square units.

Example 2: Find the area of an ellipse with equation 4x^2 + 9y^2 = 36.

Solution: We can start by rearranging the equation into standard form by dividing both sides by 36:

x^2/9 + y^2/4 = 1

This tells us that the semi-major axis length is 3 and the semi-minor axis length is 2. Now we can use the formula for the area of an ellipse, A = πab, where a = 3 and b = 2:

A = π(3)(2) = 6π

Therefore, the area of the ellipse is 6π square units.

Example 3: Find the area of the ellipse with center (2,3), semi-major axis length of 6, and semi-minor axis length of 4.

Solution: We can start by using the standard form of an ellipse centered at (h,k):

((x-h)²/a²) + ((y-k)²/b²) = 1

we get:

((x-2)²/6²) + ((y-3)²/4²) = 1

Now we can use the formula for the area of an ellipse, A = πab, where a = 6 and b = 4:

A = π(6)(4) = 24π

Therefore, the area of the ellipse is 24π square units.

### FAQs on Area of Ellipse

Q: What is an ellipse?

A: An ellipse is a geometric shape that looks like a stretched-out circle. It is defined as the set of all points in a plane such that the sum of the distances from two fixed points, called foci, is constant.

Q: What is the formula for the area of an ellipse?

A: The formula for the area of an ellipse is A = πab, where a and b are the lengths of the semi-major and semi-minor axes, respectively.

Q: How do you find the area of an ellipse?

A: To find the area of an ellipse, you need to know the lengths of the semi-major and semi-minor axes.

Then, you can use the formula A = πab to calculate the area.

Q: What is the difference between the major axis and minor axis of an ellipse?

A: The major axis of an ellipse is the longer of the two axes and lies along the x-axis or the y-axis. The minor axis is the shorter of the two axes and is perpendicular to the major axis.

Q: Can the area of an ellipse be negative?

A: No, the area of an ellipse cannot be negative.

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