Determining the area of an elliptical segment can seem complex, but with the right formula and step-by-step approach, it becomes manageable. This article will guide you through calculating the area using the given formula.
The Formula for the Area of an Elliptical Segment
The area \( A \) of an elliptical segment is given by:
\[ A = a \cdot b \cdot \cos^{-1}\left(1 - \frac{h}{a}\right) - a \cdot b \cdot \left(1 - \frac{h}{a}\right) \cdot \sqrt{\frac{2 \cdot h}{a} - \frac{h^2}{a^2}} \]
Where:
- \( a \) is the longer axis of the ellipse.
- \( b \) is the shorter axis of the ellipse.
- \( h \) is the height of the segment.
Explanation of the Formula
1. \( \cos^{-1}\left(1 - \frac{h}{a}\right) \): This term calculates the angle subtended by the segment at the center of the ellipse.
2. \( 1 - \frac{h}{a} \): This is the normalized height of the segment, representing how high the segment is relative to the semi-major axis.
3. \( \sqrt{\frac{2 \cdot h}{a} - \frac{h^2}{a^2}} \): This term adjusts for the elliptical shape, considering both the height and the semi-major axis.
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have an ellipse with a longer axis \( a = 10 \) units, a shorter axis \( b = 6 \) units, and a segment height \( h = 3 \) units. We want to find the area of the elliptical segment.
Step 1: Identify the Given Values
Given:
- Longer axis \( a = 10 \) units
- Shorter axis \( b = 6 \) units
- Segment height \( h = 3 \) units
Step 2: Substitute the Given Values into the Formula
\[ A = 10 \cdot 6 \cdot \cos^{-1}\left(1 - \frac{3}{10}\right) - 10 \cdot 6 \cdot \left(1 - \frac{3}{10}\right) \cdot \sqrt{\frac{2 \cdot 3}{10} - \frac{3^2}{10^2}} \]
Step 3: Simplify Inside the Parentheses
First, calculate \( 1 - \frac{h}{a} \):
\[ 1 - \frac{3}{10} = 0.7 \]
Then, calculate \( \cos^{-1}(0.7) \):
\[ \cos^{-1}(0.7) \approx 45.57^\circ \text{ (in radians, this is approximately 0.7954)} \]
Next, calculate \( \sqrt{\frac{2 \cdot h}{a} - \frac{h^2}{a^2}} \):
\[ \sqrt{\frac{2 \cdot 3}{10} - \frac{3^2}{10^2}} = \sqrt{\frac{6}{10} - \frac{9}{100}} = \sqrt{0.6 - 0.09} = \sqrt{0.51} \approx 0.7141 \]
Step 4: Calculate Each Term Separately
\[ 10 \cdot 6 \cdot 0.7954 = 47.724 \]
\[ 10 \cdot 6 \cdot 0.7 \cdot 0.7141 = 10 \cdot 6 \cdot 0.49987 \approx 29.9922 \]
Step 5: Subtract the Second Term from the First Term
\[ A = 47.724 - 29.9922 \approx 17.7318 \]
Final Value
For an ellipse with a longer axis of 10 units, a shorter axis of 6 units, and a segment height of 3 units, the area of the elliptical segment is approximately 17.73 square units.
