spherical cap

To calculate the area of a spherical cap given the arc length on a spherical planet, you first need to understand the relationship between the arc length, the radius of the sphere, and the central angle that subtends the arc. The area of a spherical cap can then be expressed in terms of these quantities.

Given: §

• $V$: Arc length
• $r$: Radius of the sphere (e.g., Earth’s radius)
• $\theta$: Central angle in radians

Relationships: §

1. Arc Length and Central Angle: The arc length $V$ is related to the radius $r$ and the central angle $\theta$ by the formula $V=r\cdot \theta$.

2. Central Angle and Spherical Cap Area: The area $A$ of a spherical cap is given by $A=2\pi rh$, where $h$ is the height of the cap. The height of the cap can be related to the central angle $\theta$ by the formula $h=r(1-\cos (\theta /2))$.

Calculating the Area of the Spherical Cap: §

1. Find the central angle $\theta$ using the arc length: $\theta =\frac{V}{r}$.

2. Calculate the height of the cap $h$ using $h=r(1-\cos (\theta /2))$.

3. Finally, calculate the area of the cap $A$ using $A=2\pi rh$.

Let’s express this in a formula:

$A=2\pi r\left[r(1-\cos (\frac{V}{2r}))\right]$

This formula allows you to calculate the area of a spherical cap given the arc length $V$ and the radius of the sphere $r$.