The surface area of an ellipsoid cap can be found using the approximate formula:

$S \approx 2\pi a b \left( 1 - \frac{c - h}{c} \right)^{\frac{p}{2}}$

where:

• $a, b, c$ are the semi-axes of the ellipsoid,

• $h$ is the cap height,

• $p$ is an exponent that depends on the shape of the ellipsoid, commonly approximated as:

  $p \approx 1.6075$

Given values:

• $a = 8$
• $b = 6$
• $c = 5$
• $h = 3$

Now, let's compute the surface area. math

Given values

a = 8 # Semi-axis along x b = 6 # Semi-axis along y c = 5 # Semi-axis along z h = 3 # Cap height

Exponent p approximation

p = 1.6075

Compute the surface area of the ellipsoid cap

S = 2 * math.pi * a * b * ((1 - (c - h) / c) ** (p / 2)) SThe approximate surface area of the ellipsoid cap is 200.04 square units.

Would you like a more detailed explanation or a step-by-step breakdown?

Here are five related questions:

1. How do you derive the formula for the surface area of an ellipsoid cap?
2. How does the formula change if the cap height is very small?
3. What is the total surface area of the full ellipsoid?
4. How can this formula be applied in real-world scenarios, such as fluid dynamics?
5. How does the shape of the cap change when the ellipsoid becomes more elongated?

Tip: The formula used here is an approximation. For a more precise result, an integral approach using elliptic integrals is needed.