Find the area and perimeter of an ellipse having a major axis of 10 inches and minor segment of 4 inches

To find the area and perimeter of an ellipse, we use the following formulas:

### 1. **Area of an ellipse:**
The area \( A \) of an ellipse is given by:
\[
A = \pi \times a \times b
\]
Where:
- \( a \) is the semi-major axis (half of the major axis),
- \( b \) is the semi-minor axis (half of the minor axis).

Given:
- Major axis = 10 inches, so \( a = \frac{10}{2} = 5 \) inches,
- Minor axis = 4 inches, so \( b = \frac{4}{2} = 2 \) inches.

Now, calculate the area:
\[
A = \pi \times 5 \times 2 = 10\pi \, \text{square inches}.
\]
So, the area is approximately:
\[
A \approx 31.42 \, \text{square inches}.
\]

### 2. **Perimeter of an ellipse:**
The exact perimeter \( P \) of an ellipse does not have a simple formula, but it can be approximated using **Ramanujan's approximation**:
\[
P \approx \pi \left( 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right)
\]

Substituting \( a = 5 \) and \( b = 2 \) into the formula:
\[
P \approx \pi \left( 3(5 + 2) - \sqrt{(3 \times 5 + 2)(5 + 3 \times 2)} \right)
\]
\[
P \approx \pi \left( 21 - \sqrt{(15 + 2)(5 + 6)} \right)
\]
\[
P \approx \pi \left( 21 - \sqrt{17 \times 11} \right)
\]
\[
P \approx \pi \left( 21 - \sqrt{187} \right)
\]
\[
P \approx \pi \left( 21 - 13.67 \right)
\]
\[
P \approx \pi \times 7.33 \approx 23.02 \, \text{inches}.
\]

### Final results:
- **Area**: \( 10\pi \approx 31.42 \, \text{square inches} \),
- **Perimeter**: \( \approx 23.02 \, \text{inches} \).

Would you like further details or explanations?

Here are 5 related questions:
1. What is the eccentricity of the given ellipse?
2. How would the area change if the major axis was doubled?
3. What is the focal length of the ellipse with these dimensions?
4. How does the perimeter approximation change for a circular shape?
5. Can you derive Ramanujan's approximation for the perimeter?

**Tip**: For highly eccentric ellipses, Ramanujan's approximation becomes less accurate, and numerical integration may be a better approach.

Find the area and perimeter of an ellipse having a major axis of 10 inches and minor segment of 4 inches.

Learn how to calculate the area and perimeter of an ellipse with a major axis of 10 inches and a minor axis of 4 inches. This guide uses the formula for ellipse area and Ramanujan's approximation for the perimeter. Suitable for high school geometry students in Grades 9-12.

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