Step 1: Unveil the Formula
The formula for the area (\(A\)) of an ellipse with semi-major axis (\(a\)) and semi-minor axis (\(b\)) lengths is:
\[ A = \pi a b \]
Step 2: Setting the Stage
Let's embark on an example. Suppose we have an ellipse where \(b\) is twice the length of \(a\).
Step 3: Defining Axes Relationship
Given that \(b\) is twice \(a\), we express it as:
\[ b = 2a \]
Step 4: Bridging to the Formula
With \(b = 2a\), we substitute into the area formula:
\[ A = \pi a (2a) \]
Step 5: Solving for \(a\)
\[ A = 2\pi a^2 \]
Solve for \(a\):
\[ a^2 = \frac{A}{2\pi} \]
\[ a = \sqrt{\frac{A}{2\pi}} \]
Step 6: Computing \(b\)
Now, we determine \(b\) using the relationship \(b = 2a\):
\[ b = 2 \times \sqrt{\frac{A}{2\pi}} \]
Step 7: Final Calculation
Let's calculate a concrete example. Say the area \(A\) of the ellipse is 50 square units.
\[ a = \sqrt{\frac{50}{2\pi}} \]
\[ a \approx \sqrt{\frac{25}{\pi}} \]
\[ a \approx 2.8209 \, \text{units} \]
Now for \(b\):
\[ b = 2 \times 2.8209 \]
\[ b \approx 5.6419 \, \text{units} \]
