Capsule Calculator

What is a Capsule?

A capsule is a three-dimensional geometric shape formed by combining a cylinder with two hemispheres at its ends. It resembles a pill or a space capsule, hence its name. Capsules are commonly found in pharmaceutical products, space exploration vehicles, and various industrial applications.

How to Calculate Capsule Properties

To fully understand a capsule, we need to calculate several key properties: its radius, height, volume, surface area, and lateral surface area. Each of these properties provides unique information about the capsule's dimensions and characteristics.

Formulas

Here are the essential formulas for a capsule:

1. Volume (V):

\[ V = \frac{4}{3}\pi r^3 + \pi r^2 h \]

2. Surface Area (A):

\[ A = 4\pi r^2 + 2\pi r h \]

3. Lateral Surface Area (L):

\[ L = 2\pi r h \]

4. Circumference (C):

\[ C = 2\pi r \]

Where:

• \(r\) is the radius of the capsule
• \(h\) is the height of the cylindrical part of the capsule
• \(\pi\) is the mathematical constant pi (approximately 3.14159)

Calculation Steps

1. Determine the radius (r) and height (h) of the capsule
2. Calculate the volume using \(V = \frac{4}{3}\pi r^3 + \pi r^2 h\)
3. Calculate the surface area using \(A = 4\pi r^2 + 2\pi r h\)
4. Calculate the lateral surface area using \(L = 2\pi r h\)
5. Calculate the circumference using \(C = 2\pi r\)

Example and Visual Representation

Let's calculate the properties of a capsule with a radius of 2 units and a height of 6 units:

1. Given: \(r = 2\) units, \(h = 6\) units
2. Volume: \(V = \frac{4}{3}\pi r^3 + \pi r^2 h = \frac{4}{3}\pi (2^3) + \pi (2^2)(6) \approx 100.53\) cubic units
3. Surface Area: \(A = 4\pi r^2 + 2\pi r h = 4\pi (2^2) + 2\pi (2)(6) \approx 100.53\) square units
4. Lateral Surface Area: \(L = 2\pi r h = 2\pi (2)(6) \approx 75.40\) square units
5. Circumference: \(C = 2\pi r = 2\pi (2) \approx 12.57\) units

Here's a visual representation of this capsule:

In this diagram, you can see a 2D representation of our capsule with radius 2 units and height 6 units. The blue surface represents the capsule's surface, the red line shows the radius, and the green line shows the height. The radius (r), height (h), volume (V), surface area (A), lateral surface area (L), and circumference (C) are labeled.