# Volume and Surface Area of a Cylinder Formula

In geometry, a cylinder is a three-dimensional shape with two parallel circular discs or bases separated by a certain distance. At a fixed distance from the center, the two circular bases are connected by a curved surface. The distance between the two parallel circular bases of the cylinder is referred to as the height of the cylinder. A cylinder has a lateral surface and two bases that are parallel and identical. Some of the real-life examples of a cylinder are canned food containers, cold-drink cans, gas cylinders, toilet paper rolls, etc.

### Definition and types of cylinders

In geometry, a cylinder is a three-dimensional shape with two circular bases connected by a lateral surface known as the curved surface. The distance between the two parallel circular bases of the cylinder is referred to as the height (h) of the cylinder and the line connecting the centers of the two circular bases is the axis of the cylinder. The radius (r) of a cylinder is the distance from the center to the outer boundary of a cylinder. A cylinder looks like a circle from the top view and a rectangle from the side view.

Cylinders are classified into various types, such as right circular cylinders, Oblique cylinders, Elliptical cylinders, and cylindrical shells or hollow cylinders.

**Right Circular Cylinder:**A right circular cylinder is a cylinder with the axis of the cylinder perpendicular to the center of the base.**Oblique cylinders:**An oblique cylinder is a cylinder with the axis of the cylinder not perpendicular to the center of the base, i.e., the axis makes an angle other than a right angle with the center of the base.**Elliptical cylinders:**An elliptical cylinder is a cylinder having elliptical-shaped bases.**Cylindrical shells or hollow cylinders:**A hollow cylinder is made up of two right-circular cylinders that are bound inside one another. It is void from the inside and has a difference between the inner and outer radii. The point of the axis is perpendicular to the central base and common to both cylinders. Some real-life examples of hollow cylinders are hollow pipes, toilet paper rolls, etc.

### Formulae for Cylinder

Like every other three-dimensional geometric shape, even a cylinder has two major formulae, i.e., surface area and volume. A cylinder has two kinds of surface areas: the curved surface area or the lateral surface area, and the total surface area.

So, the three major formulae related to a cylinder are,

- Curved surface area or Lateral surface area
- Total surface area
- Volume

**Curved surface area or Lateral surface area**

The curved surface area, or lateral surface area, of a cylinder, is the space enclosed between the two parallel circular bases.

The formula for the curved surface area, or lateral surface area, of a cylinder, is given as,

Curved Surface Area (CSA) = Circumference × Height

CSA = 2πr × h = 2πrh square units

Curved Surface Area (CSA) = 2πrh square unitsWhere

r is the radius and

h is the height of the cylinder.

**Total surface area**

The total surface area of a cylinder is the total area enclosed by a cylinder, including its bases. A cylinder has a lateral surface and two bases that are parallel and identical. Therefore, the total surface area of a cylinder is the sum of the area of the curved surface or lateral surface and the areas of the two circular bases.

We know that,

The curved Surface Area (CSA) = (2 π r h) square units

Area of a Circle = πr^{2} square units

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

TSA = 2πrh + 2πr^{2} = 2πr(h + r) square units

Thus, the formula for the total surface area of a cylinder is given as,

Total surface area of the cylinder = [2πr(h + r)] square unitsWhere

r is the radius and

h is the height of the cylinder.

**Volume of Cylinder**

The volume of a cylinder is the density or amount of space occupied by the cylinder. Let us assume that a cylindrical-shaped container is filled with refined oil. Now, to calculate the amount of oil, we need to determine the volume of the cylindrical-shaped container.

Now, the volume of a cylinder = Area of a circle × height

Volume (V) = πr^{2} × h cubic units

Thus, the formula for the volume of a cylinder is given as,

Volume of a cylinder = (πr^{2}h) cubic unitsWhere

r is the radius and

h is the height of the cylinder.

### Sample Problems

**Problem 1: Determine the curved surface area of the cylinder with a radius of 8 inches and a height of 15 inches.**

**Solution:**

Given,

Radius = 7 inches and

The height of the cylinder = 15 inches.

We have,

The curved surface area of the cylinder = (2πrh) square units

= 2 × (22/7) × 8 × 15

= 754.285 sq. in

Hence, the curved surface area of the cylinder is 754.285 sq. in.

**Problem 2: Calculate the volume of a cylindrical-shaped water container that has a height of 18 cm and a diameter of 12 cm.**

**Solution:**

Given,

The height of the cylinder = 18 cm

Diameter = 12 cm

⇒ 2 × radius = 12 cm

⇒ r = 6 cm

We have,

The volume of a cylinder = (π r

^{2}h) cubic units⇒ V = (22/7) × (6)

^{2 }× 18⇒ V = 2,034.72 cm

^{3}Hence, the volume of the cylindrical-shaped water container is 2034.72 cm

^{3}.

**Problem 3: Determine the total surface area of the cylinder with a radius of 6 cm and a height of 12 inches.**

**Solution:**

Given,

Radius = 6 cm and

The height of the cylinder = 12 cm

We have,

The total surface area of the cylinder = [2πr(h + r)] square units

= 2 × (22/7) × (6) × (6 + 12)

= 37.714 × 18 = 678.857 sq. cm.

Therefore, the total surface area of the cylinder is 678.857 sq. cm.

**Problem 4: Determine the height of the cylinder if its volume is 625 cubic units and its radius is 5 units.**

**Solution:**

Given,

The volume of the cylinder = 625 cubic units

Radius = 5 units

The height of the cylinder =?

We know that,

The volume of a cylinder = (π r

^{2}h) cubic units⇒ 625 = (22/7) × (5)2 × h

⇒ (22/7) × 25 × h = 625

⇒ h = (625/25) × (7/22)

⇒ h = 7.95 units

Hence, the height of the given cylinder is 7.95 units

**Problem 5: Determine the curved surface area of a cylinder if its total surface area is 770 sq. in and its radius is 7 inches.**

**Solution:**

Given,

Radius = 7 inches

The total surface area of the cylinder = 770 sq. in

We know that,

The total surface area of a cylinder = [2πr(h + r)] square units

⇒ 770 = 2 × (22/7) × (7) × (7 + h)

⇒ 2 × (22) × (7 + h) = 770

⇒ 7 + h = 17.5

⇒ h = 10 cm

Now,

The curved surface area of cylinder = 2πrh square units

= 2 × (22/7) × (7) × (10)

= 2 × (22) × (10) = 440 sq. in.

Thus, the curved surface area of the given cylinder is 440 sq. in.

**Problem 6: Find the radius of the cylinder if its curved surface area is 550 sq. cm and its height is 14 cm.**

**Solution:**

Given,

The curved surface area of the cylinder = 550 sq. cm

The height of the cylinder = 14 cm

Radius =?

We know that,

The curved surface area of the cylinder = (2πrh) square units

⇒ 550 = 2 × (22/7) × r × 14

⇒ 88r = 550 ⇒ r = 550/88

⇒ r = 6.25 cm

hence, the radius of the given cylinder is 6.25 cm.

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