Why CSA of cylinder is 2πrh?

Introduction

The CSA (cross sectional area) of a cylinder is 2πrh. But why? It seems like such an arbitrary number, and you may be wondering how anyone ever came up with it in the first place. In this blog post, we will explore the reasoning behind the CSA of a cylinder. We will also look at how to calculate it and what other properties of cylinders can be determined from its CSA. So if you’ve ever wondered why the CSA of a cylinder is 2πrh, read on!

The Area of a Cylinder

The area of a cylinder can be found by multiplying the radius of the cylinder by the height of the cylinder. This gives the formula for the area of a cylinder: A = πr2h.

The reason why the formula for the area of a cylinder is πr2h is because the surface area of a cylindrical object is composed of two circular ends (of radius r) and a rectangle wrapped around its circumference (of height h). The total surface area, then, is simply two times the sum of these three areas:

A = 2πr2 + 2πrh

The Surface Area of a Cylinder

The surface area of a cylinder is the sum of the areas of its top and bottom circles, as well as the area of its lateral (side) surface. The formula for the surface area of a cylinder is therefore:

Surface Area = πr2 + πr2 + 2πrh

= 2πr(r + h)

With r being the radius of the cylinder’s base and h being its height.

Volume of a Cylinder

If you have ever wondered why the formula for the volume of a cylinder is πr^2h, then read on! In this article, we will explore the derivation of this well-known result.

We will start with some simple observations. First, notice that a cylinder can be thought of as composed of many thin disks (or slices). Each slice has a thickness of dr and an radius of r. The volume of each slice is therefore 2πr*dr. If we add up all of these volumes, we get the total volume of the cylinder:

V = ∫0h 2πr*dr

Now let’s take a closer look at that integral. Recall that the definite integral can be thought of as a summation: it is the sum of all infinitesimal slices. In our case, we are summing up all of the cylindrical slices from r=0 to r=h. But notice that each slice has a different radius! So how can we write this as a single summation?

The key is to realize that the radius is changing by an amount dr each time. That is, we can rewrite our integral as follows:

V = ∫0h 2π(r+dr)*dr

Now we can see that this is simply a summation, where each term has a radius that is increased by dr from the previous term. We can thus write:

Why Is the CSA of a Cylinder 2πrh?

The CSA of a cylinder is 2πrh because the base of the cylinder is a circle. The circumference of a circle is 2πr, so the area of the base is πr². The height of the cylinder is h, so the CSA is πr² + 2πrh.
The CSA of a cylinder can be determined by a few different methods. The first and most intuitive method is to simply slice the cylinder in half and look at the resulting cross section. As you can see in the figure below, the CSA of the resulting shape is a circle with radius r.

![Cross section of a cylinder](cylinder-cross-section.png)

The area of a circle is πr², so the CSA of the cylinder is 2πr².

Another way to determine the CSA of a cylinder is to think about how much surface area would be required to wrap around the cylinder. If we unwrapped the cylinder, it would look like a rectangle with dimensions 2πr by h (see figure below).

![Unwrapped cylinder](cylinder-unwrapped.png)

The surface area of a rectangle is lw, so the CSA of the cylinder is 2πrh.

Conclusion

While the formula for the CSA of a cylinder may seem confusing at first, it actually makes a lot of sense when you think about it. The radius is responsible for determining the size of the circle, and since there are two circles (one at each end), we need to multiply by two. The height determines how tall the cylinder is, so we need to multiply that by the circumference of the circle (which is 2πr). All in all, the formula makes perfect sense once you think about it!

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