Spherical Cap Calculator: Volume, Surface Area & More

What is a Spherical Cap?

A spherical cap is a portion of a sphere cut off by a plane. It's like slicing off the top of a ball, leaving you with a dome-shaped object. Spherical caps are common in geometry and have practical applications in architecture, engineering, and even in nature, such as water droplets on a surface.

How to Calculate Spherical Cap Properties

To fully understand a spherical cap, we need to calculate several key properties: its height, base radius, surface area, volume, and arc length. Each of these properties provides unique information about the cap's size and shape.

Formulas

Here are the essential formulas for a spherical cap:

1. Height (h):

\[ h = R \pm \sqrt{R^2 - r^2} \]

2. Base Radius (r):

\[ r = \sqrt{2Rh - h^2} \]

3. Surface Area (A):

Total Surface Area: \[ A_{total} = \pi(r^2 + h^2) \]

Curved Surface Area: \[ A_{curved} = 2\pi Rh \]

4. Volume (V):

\[ V = \frac{1}{3}\pi h^2(3R - h) \]

5. Arc Length (L):

\[ L = 2\pi R \arcsin(\frac{r}{R}) \]

Where:

• \(R\) is the radius of the sphere
• \(r\) is the radius of the base of the cap
• \(h\) is the height of the cap
• \(\pi\) (pi) is approximately 3.14159

Calculation Steps

1. Determine any two of the three main parameters: R, r, or h
2. Calculate the third parameter using the appropriate formula
3. Calculate the total surface area using \(A_{total} = \pi(r^2 + h^2)\)
4. Calculate the curved surface area using \(A_{curved} = 2\pi Rh\)
5. Calculate the volume using \(V = \frac{1}{3}\pi h^2(3R - h)\)
6. Calculate the arc length using \(L = 2\pi R \arcsin(\frac{r}{R})\)

Example and Visual Representation

Let's calculate the properties of a spherical cap with a sphere radius (R) of 10 units and a base radius (r) of 6 units:

1. Height: \(h = 10 - \sqrt{10^2 - 6^2} = 1.87\) units
2. Total Surface Area: \(A_{total} = \pi(6^2 + 1.87^2) = 119.37\) square units
3. Curved Surface Area: \(A_{curved} = 2\pi (10)(1.87) = 117.50\) square units
4. Volume: \(V = \frac{1}{3}\pi (1.87)^2(3(10) - 1.87) = 35.19\) cubic units
5. Arc Length: \(L = 2\pi (10) \arcsin(\frac{6}{10}) = 38.20\) units

Here's a visual representation of this spherical cap:

In this diagram, you can see our spherical cap with sphere radius R = 10 units and base radius r = 6 units. The height h, approximately 1.87 units, is the distance from the base to the top of the cap. The volume of 35.19 cubic units represents the space enclosed by the cap.