# Finding the Volume and Surface Area of a Cylinder

In geometry, a cylinder is a three-dimensional solid figure, containing two parallel circular bases joined by a curved surface, situated at a particular distance from the center of the cylinder. For instance, toilet paper rolls, plastic cold drink cans are examples of cylinders. A cylinder is characterized by two major properties, i.e., surface area and volume. The word cylinder is derived from a Latin (Cylindrus) word, meaning “roll”, “roller”, and “tumblr”.

**Definition of a cylinder**

A cylinder is a three-dimensional solid which contains two parallel bases which are held together by a curved surface, held at a fixed distance. The bases of the cylinder are normally circular in shape, similar to a circle. The bases are held together by a line segment, which is referred to as the axis. The distance from this line segment to the outer surface of the cylinder is called the radius. This can be denoted by ‘r’. The perpendicular distance between the bases of the cylinder is called the height of the cylinder, referred by ‘h’.

**Parts of a cylinder**

The cylinder is considered to be composed of 2 circles + 1 rectangle. The following image depicts the formation of cylinder:

**Types of Cylinder**

Geometry consists of four types of cylinders, namely,

**Elliptic cylinder:**A cylinder base forming an ellipse is called an elliptic cylinder.**Right circular cylinder:**The right circular cylinder contains the axes of the two parallel lines which are perpendicular to the center of the base.**Oblique cylinder:**An oblique cylinder is one whose sides lean over the base. In an oblique cylinder, the sides are not perpendicular to the center of the base. The Leaning Tower of Pisa is a real-world example of an oblique cylinder.**Right circular hollow cylinder or cylindrical shell:**Also termed as, a cylindrical shell, contains two right circular cylinders bounded by one side on the other. The point of the axis is common to the intersection and is perpendicular to the central base. Since there is some space inside the cylinder, it is hollow from the inside.

**Formulas of Cylinder**

The cylinder is associated with three formulae, finding its applications with area and volume:

- Lateral Surface Area or Curved Surface Area
- Total Surface Area
- Volume of the Cylinder

**Lateral Surface Area or Curved Surface Area of Cylinder**

The curved surface area is also termed a lateral surface area. The area formed by the curved surface of the cylinder i.e. space occupied between the two parallel circular bases is known as CSA. The formula for CSA is given as:

Curved Surface Area (CSA) = 2Ï€rh square unitsHere, â€˜hâ€™ is the height and â€˜râ€™ is the radius

**Total Surface Area of Cylinder**

So, in order to find out the total surface area of a cylinder, we calculate the curved surface area and the area of two circles.

The total surface area of the cylinder is defined as the total area occupied by it. A cylinder consists of two circles along with a curved sheet. The total surface area of a cylinder can be calculated by the combination of curved surface area and the area of two circles.

Curved Surface Area(CSA) = Circumference of the Circle Ã— Height

C.S.A = 2r Ã— h

Area of a Circle = Ï€r^{2}

Total Surface Area (TSA) = Curved Surface Area + 2(Area of a circle)

We know,

Curved Surface Area = 2Ï€rh

Area of circle = Ï€r^{2}

Total Surface Area (T.S.A) = 2Ï€rh + 2Ï€r^{2}= 2Ï€r(h+r) square units.Since, there are two circles, the calculation is performed by multiplying with 2.

where, h is the height and r is the radius of the cylinder.

**Volume of Cylinder**

The volume of the cylinder is referred to as the density or amount of space it occupies.

We have,

Volume of a cylinder = Area of a circle Ã— height

Since, we have an area of a circle = Ï€r^{2}

Volume = Ï€r^{2}Ã— hTherefore, V = Ï€r

^{2}h cubic unitswhere h is the height and r is the radius of the cylinder.

### Sample Problems

**Question 1: Compute the total surface area of the cylinder, with a radius of 5cm and height of 10cm?**

**Solution: **

Since, we know,

Total surface area of a cylinder, A = 2Ï€r(r+h) square units

Therefore, A = 2Ï€ Ã— 5(5 + 10) = 2Ï€ Ã— 5(15)

= 2Ï€ Ã— 75 = 150 Ã— 3.14

= 471 cm

^{2}

**Question 2: What is the volume of a cylindrical shape water container, that has a height of 7cm and diameter of 10cm?**

**Solution: **

Given,

Diameter of the container = 10cm

Thus, the radius of the container = 10/2 = 5cm

Height of the container = 7cm

As we know, from the formula,

Volume of a cylinder = Ï€r

^{2}h cubic units.Therefore, volume of the given container, V = Ï€ Ã— 52 Ã— 7

V = Ï€ Ã— 25 Ã— 7 = (22/7) Ã— 25 Ã— 7 = 22 Ã— 25

V = 550 cm

^{3}

**Question 3: Alex wants to purchase a cylindrical can with a** **radius equivalent to 5 inches. The can contains 1 gallon of oil. Find the height of the** **cylinder. **

**Solution:**

Volume V is given by= 1 gallon

1 gallon= 231 cubic inches

Radius r = 5 inches

Volume f the cylinder is given by,

V = Ï€r

^{2}h231 = 22/7 Ã— (5)

^{2}Ã— h(231 Ã— 7)/(22 Ã— 25) = h

h = 2.94 inches.

Therefore, the height is equivalent to 2.94 inches.

**Question 4. A water tank has a radius of 40 inches and a height of 150 inches. Find the area. **

**Solution:**

Water tank is cylindrical in nature.

Total Surface Area of a cylinder is given by, 2Ï€r(h+r)

TSA = 2 Ã— 22/7 Ã— 40(150 + 40)

TSA = 2 Ã— 22/7 Ã— 7600

TSA = 47,771.42 sq.inches

Area = 47,771.42 sq.inches.

**Question 5. Find the volume of the cylinder having a radius of 5 units and a height of 8 units?**

**Solution:**

We have,

Radius,r = 5 units

Height,h = 8 units

Volume of the cylinder, V = Ï€r

^{2}h cubic units.V = (22/7) Ã— 5

^{2}Ã— 8V = 22/7 Ã— 25 Ã— 8

V= 628.57 Cubic units.

Hence, the volume of the cylinder is 628.57 cubic units.

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