**3-D Cylinder & its Formulas :**

A** cylinder** is one of the common **shapes** we see in everyday life. It is **three dimensional(3-D)**. It has **two** parallel **faces** which are called as **bases** or ends of the **cylinder.**

A **curved surface** joins these **two bases**. Therefore, a **cylinder area**, more predominantly referred as **surface area** of **cylinder**, includes **two types of areas. **

**One** is the **area** of the **bases**, called as **base areas**, and the other is the **area** of the joining **curved surface**. The latter is better known as the **lateral surface of a cylinder**. In general **surface area** of **cylinder** refers to the **sum** of **lateral area** and the **areas** of the** two bases.**

Let us illustrate how the **formula** for **surface area** of a **cylinder** can be derived in a simple way.

In the above **diagram**, a **cylinder** of **radius r** and **height h** is shown on the left.

Suppose the** cylinder** (considering it to be hollow) is ripped along the dotted line shown and opened on both the directions, we can see a **rectangle formed** as shown on the right. The **length** of the **rectangle** is **equal** to the **circumference** of the **end circles **

which is **2πr**,

because the entire **circumference** is opened out .The** height** of the **rectangle** remains **same** as the **height** of the **cylinder** as **h**.

It can be seen as obvious that the **area** of the **curved surface** of the **cylinder** is same as the **area** of the **rectangle**, which is

**2πr* h = 2πrh**.

Therefore, the **lateral surface area** of a **cylinder** is given by

**L = 2πrh**.

The **total surface** area is the **sum** of the **lateral surface area** and the **areas of the two bases** which are **circular**.

The **areas** of the **two bases** are the **areas two circles** which is

** πr^2 + πr^2 = 2πr^2.**

Therefore, the** total surface area** of the **cylinder** is,

** (2πr^2 + 2πrh).**

The **same** can be simplified in **factored form** as

** 2πr(r + h).**

Now let us look into the significance of **surface** **area** and **lateral area** of a **cylinder**. The **surface area** tells us the **amount of surface**, in turn, the **area** of the sheet metal to that is needed to fabricate a **cylinder**.

In many cases, **cylindrical** **shapes** in large sizes without ends are used as shells (think of water or any fluid pipe lines). Therefore, in such cases, only the** lateral surface areas** of **cylinders** play the role.

There are some more **interesting facts** about the **cylinder areas** and **cylinder volumes**. Suppose you are asked to design a **hollow cylinder** for a given **volume** that involves **minimum cost** to fabricate. That is, to find the **dimensions** with least **surface area.**

** Mathematically** we can **prove** that for a **fixed volume** of a **cylinder**, the **least surface area** can be achieved when **h = 2r**, that is when the **height** of the **cylinder equals** the **diameter** of the **cylinder**.

Same way, for a given **surface area** of a **cylinder**, **maximum** **volume** can be obtained by making the **height** and **diameter** of the **cylinder congruent**.