Area of an ellipse calculator

Definition: What is an ellipse?

An ellipse is a fundamental geometric shape that appears in various fields, from mathematics and physics to engineering and astronomy. Formally, an ellipse is defined as the set of all points in a plane such that the sum of their distances from two fixed points, called the foci, is constant. This unique property distinguishes the ellipse from other conic sections such as circles, parabolas, and hyperbolas.

Mathematically, an ellipse can be represented by its standard equation:

Understanding the components of an ellipse

Major axis and minor axis: The longest and shortest diameters of the ellipse, respectively. The major axis passes through both foci, while the minor axis is perpendicular to it.

Foci (singular: focus): These are two distinct points inside the ellipse. The constant sum of distances to the foci defines the shape.

Center: The midpoint of both the major and minor axes, serving as the symmetry point of the ellipse.

How to calculate the area of an ellipse?

Calculating the area of an ellipse is straightforward when you know the lengths of its semi-major axis (a) and semi-minor axis (b). The formula for the area of an ellipse is derived from its geometric properties:

Step-by-Step calculation

Measure the axes

• Determine the length of the semi-major axis (a).
• Determine the length of the semi-minor axis (b).

Apply the formula

• Multiply π by the product of a and b.

Compute the area

• Use a calculator to perform the multiplication for precise results.

Math problems

1. Problem 1

   Given: Semi-major axis \(a = 5\) cm, semi-minor axis \(b = 3\) cm.

   Formula for area of an ellipse: \(A = \pi \times a \times b\)

   • Step 1: Write the formula: \(A = \pi \times a \times b\).
   • Step 2: Substitute the values: \(A = \pi \times 5 \times 3\).
   • Step 3: Multiply the numbers first: \(5 \times 3 = 15\).
   • Step 4: Now multiply by \(\pi \approx 3.14\): \(A = 3.14 \times 15\).
   • Step 5: \(3.14 \times 15 = 47.1\).

   Answer: The area of the ellipse is 47.1 cm².

2. Problem 2

   Given: Semi-major axis \(a = 10\) m, semi-minor axis \(b = 4\) m.

   Formula: \(A = \pi \times a \times b\)

   • Step 1: \(A = \pi \times a \times b\).
   • Step 2: Substitute: \(A = \pi \times 10 \times 4\).
   • Step 3: Multiply the axes: \(10 \times 4 = 40\).
   • Step 4: Multiply by \(\pi \approx 3.14\): \(A = 3.14 \times 40\).
   • Step 5: \(3.14 \times 40 = 125.6\).

   Answer: The area of the ellipse is 125.6 m².

3. Problem 3

   Given: Semi-major axis \(a = 6\) cm, semi-minor axis \(b = 6\) cm.

   Note: When \(a = b\), the ellipse is a circle, but the same formula works.

   Formula: \(A = \pi \times a \times b\)

   • Step 1: \(A = \pi \times a \times b\).
   • Step 2: Substitute: \(A = \pi \times 6 \times 6\).
   • Step 3: Multiply the axes: \(6 \times 6 = 36\).
   • Step 4: Multiply by \(\pi \approx 3.14\): \(A = 3.14 \times 36\).
   • Step 5: \(3.14 \times 36 = 113.04\).

   Answer: The area of the ellipse is 113.04 cm².

4. Problem 4

   Given: Semi-major axis \(a = 8.2\) m, semi-minor axis \(b = 3.5\) m.

   Formula: \(A = \pi \times a \times b\)

   • Step 1: \(A = \pi \times a \times b\).
   • Step 2: Substitute: \(A = \pi \times 8.2 \times 3.5\).
   • Step 3: Multiply the axes first: \(8.2 \times 3.5 = 28.7\).
   • Step 4: Now multiply by \(\pi \approx 3.14\): \(A = 3.14 \times 28.7\).
   • Step 5: \(3.14 \times 28.7 \approx 90.12\).

   Answer: The area of the ellipse is 90.12 m² (rounded to two decimal places).

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