A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface. The total area occupied by the surface of a cylindrical shape in a three-dimensional space is known as its surface area. To recall, a cylinder is a three-dimensional structure having circular bases which are parallel to each other. It does not have any vertices. You can imagine a cylinder as a collection of circular disks stacked on top of each other.

We calculate the surface area of a cylinder in two ways: total surface area (TSA) and curved/lateral surface area (CSA). The cylinder surface area is represented in square units such as cm², m², and so on. We will discuss the definition of surface area of cylinder along with formulas to calculate it. We have also provided an online calculator to find the surface area of any cylinder using its radius and height.

What is a Cylinder? Definition & Equation

A cylinder is defined as a solid geometrical figure with straight parallel sides and a circular or oval cross-section. You can understand a cylinder as multiple circular disks stacked on top of each other. The radius of the disk is the same as the radius of the newly-formed cylinder. The number of disks determines its height.

Geometric definition of a cylinder: A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.

Cylinder Equation: Graphically, the general equation of a cylinder is given by:

$${(x – a)^2} + {(y – a)^2} = {r^2}$$

where x, y, and z are spatial coordinates, a is the distance of the center of the cylinder from the origin and r is the radius. If the center of the cylinder lies at (overlaps with) origin, then the equation of a cylinder becomes:

Surface Area of Cylinder Online Calculator

Below, you can find the simple online calculator to find the curved surface area and total surface area of cylinder. Just enter the value of the radius and height of the cylinder, then click on “Calculate” button to get the corresponding value of CSA and TSA.

Radius Height Curved Surface Area (CSA) Total Surface Area (TSA)

Surface Area of Cylinder Formula

The area of a cylinder is the total region covered by the surface of the cylinder in three-dimensional space. The cylinder area is equal to the sum of the area of two circular bases and curved surface area.

The total area of a cylinder is based on:

1. Curved Surface Area (CSA)
2. Base Area

1. Curved Surface Area (CSA): The curved surface area of a cylinder (CSA) is defined as the area of the curved surface of any given cylinder having base radius ‘r’, and height ‘h’, It is also termed as Lateral surface area (LSA). The formula for a curved area or lateral area is given by:

$$CSA(LSA) = 2 \times \pi \times r \times h$$

2. Base Area of Cylinder: The base of the cylinder is in a circular shape. Hence, by the formula of area of the circle, we have:
$$Base\,Area\,of\,Cylinder = 2 \times \pi {r^2}$$

Total Surface Area of Cylinder

The total surface area of a cylinder is given as:

Total Surface Area = Curved Surface Area + Base Area

$$TSA = 2\pi rh + 2\pi {r^2} = 2\pi r(h + r)$$

Surface Area of Cylinder Formula Derivation

To derive the formula for surface area of cylinder, consider a cylinder with radius r and height h. Now unroll the cylinder just like you unroll a curled up mat. Refer to the image below:

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After complete unroll, the cylinder will look like this:

From the figure above, we can see that the curved or lateral surface area can be calculated by finding the area of rectangle with length 2?r and width h. So,

Curved Surface Area = 2?r * h = 2?rh

Also, if we want to calculate the total surface area, we will add the area of two circular bases (top and bottom) with radius r. Hence,

Total Surface Area = 2?rh + 2?r²

Surface Area of Cylinder Solved Examples

Here are some of the solved problems on surface area of a cylinder.

Question 1: Calculate the cost required to paint a container which is in the shape of a right circular cylinder having a base radius of 7 m and height 13 m. If the painting cost of the container is ₹2/m². (Take π = 22/7).

Solution: Total surface area of container = 2πr (h + r)= 2 x 22/7 x 7 x 20 = 880 m²
Total cost of painting the container = 2 × 880 = ₹1760

Question 2: Find the total surface area of a container in cylindrical shape whose diameter is 28 cm and height is 15 cm.

Solution: Given, diameter = 28 cm, so radius = 28/2 = 14 cm and height = 15 cm
By the formula of total surface area, we know;
TSA = 2πr (h + r)
= 2x 22/7 x 14 x (15 + 14)
= 2 x 22 x 2 x 29
= 2552 sq.cm
Hence, the total surface area of container is 2552 cm²

FAQs

Here are some of the frequently asked questions on surface area of cylinder.

Q1: How do I calculate the surface area of a cylinder? A: To calculate the surface area of a cylinder, we have to find the lateral surface area and area of circular bases of cylinder. Then add the two areas to get the total surface area.

Q2: What is the relation between curved surface area and total surface area of a cylinder? A: Total surface area of cylinder is the sum of curved surface area and area of the two circular bases.

Q3: What is the formula to find the surface area of a cylinder? A: The formula to calculate the total surface area of a cylinder is 2?rh + 2?r2.

Q4: What is a cylinder? A: A cylinder is a three-dimensional shape having two circular bases in parallel to each other joined by a curved surface.

We hope this detailed article on surface area of cylinder is helpful to you. If you have any queries, ping us through the comment box below and we will get back to you as soon as possible.