A capsule is a three-dimensional geometric shape that resembles a cylindrical tube with two hemispherical ends. The volume of a capsule represents the amount of space enclosed by its surface. To calculate the volume of a capsule, we consider the combined volumes of the two hemispheres and the cylinder in the middle.
First, we find the volume of each hemisphere. The formula for the volume of a hemisphere is (2/3)πr³, where π (pi) is a mathematical constant approximately equal to 3.14159, and r represents the radius of the hemispherical end.
Next, we calculate the volume of the cylindrical portion. The formula for the volume of a cylinder is πr²h, where r is the radius of the cylinder’s base and h is its height.
Finally, we add the volumes of the two hemispheres and the cylinder to obtain the total volume of the capsule.
Let’s consider an example to better understand this concept. Suppose we have a capsule with a hemisphere radius of 2 units and a cylindrical portion with a radius of 2 units and a height of 8 units. We can calculate the volume as follows:
Volume of hemisphere = (2/3)π(2)³
Volume of cylindrical portion = π(2)²(8)
Total volume = 2 × Volume of hemisphere + Volume of cylindrical portion
Hence, the volume of the capsule with a radius of 2 units and height of 8 units is approximately 134.04129 cubic units.