Surface Area of a Cylinder Formula: Definition, Formulae, Examples

Surface Area of a Cylinder Formula: A solid occupies a fixed amount of space. Solids are available in a wide range of shapes. All solids have a surface and, as a result, a surface area, as we’ve seen. Because it takes up space, every solid has volume.

A cylinder is a three-dimensional solid object made up of two circular bases joined by a curved face. We can express the curved and total surface area of a cylinder because it has a curved surface. Let’s look at cylinders and the surface area formula. Continue reading to know more.

What is a Cylinder?

A cylinder is a basic three-dimensional geometric shape with one curved lateral surface and two flat round surfaces at each end. A cylinder contains three faces: one curved face and two flat circular faces, two edges (where two faces connect), and no vertices (corners where two edges meet).

Surface Area of a Cylinder

The amount of space covered by the flat surface and the curved surface of a cylinder is known as its surface area. The cylinder’s surface area is the sum of the area of its circular bases (because the cylinder’s base is a circle) and the area of the curved surface (which is a rectangle with the cylinder’s height as length and circumference as its breadth).

The cylinder’s entire surface area is made up of two parts:

1. Curved surface area of a cylinder
2. Total surface area of a cylinder

Curved Surface Area of a Cylinder

The curved surface area is known as the area of the curved face of the cylinder. The term “curved surface area” refers to the area of a curve. The area of a cylinder minus the area of its circular bases is called the curved surface area.

In general, square units such as centimetre square, meter square, and so on are used to measure the area. A rectangular shape is found when the curved surface area of the cylinder is opened.

A cylinder has three faces, as we can see. One rectangle (when cut and made flat) and two circles. The cylinder has two circles, one at the bottom and the other at the top. These two circles are identical in size. The area of the rectangular face calculates the curved surface of the cylinder. So,

The cylinder’s circle has an area of \(\pi {r^2}.\)

\(2\pi {r^2}\) will be the area of two circles.

A cylinder’s radius is equal to the radius of its base. Now, the rectangle’s area \(A = {\rm{length \times breadth}}{\rm{.}}\)
The circumference of the circle is \(2\pi r,\) and the height is \(h.\)
The curved surface area of a cylinder\({\rm{ = 2}}\pi rh{\rm{.}}\)

Total Surface Area of a Cylinder

The total surface area of a cylinder is equal to the sum of areas of all its faces. The total surface area with radius \(r,\) and height \(h\) is equal to the sum of the curved area and circular areas of the cylinder.

\({\rm{Total}}\,{\rm{surface}}\,{\rm{area}}\left( {{\rm{TSA}}} \right) = 2\pi \times r \times h + 2\pi {r^2}\)

\({\rm{(TSA)}} = 2\pi r\left( {h + r} \right)\,{\rm{sq}}{\rm{.units}}\)

Surface Area of a Right Circular Cylinder

The right circular cylinder is a cylinder with circular bases that are parallel to each other. It has a three-dimensional appearance. The axis of the cylinder connects the centre of the two-cylinder bases. 

The total surface area of a right circular cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
The curved surface area of a right circular cylinder \(\left( {{\rm{CSA}}} \right) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\) where \(h \to {\rm{height,}}\,r \to {\rm{radius}}\)

Learn the Concepts of Surface Area of Cylinder

Derivation of Surface Area of a Cylinder

As we can see, a cylinder has three faces, two circles and a rectangle. There are two circles on the cylinder, one at the bottom and the other at the top. The size of these two circles is the same. The rectangular face is the curved surface of the cylinder.

The cylinder’s circle has an area\( = \pi {r^2}\)

The area of two circles\( = 2\pi {r^2} \ldots \ldots {\rm{(i)}}\)

A cylinder’s radius is equal to the radius of its base. 

The circumference of the circle is \(2πr,\) and the height is \(h.\)

We know, the curved surface area of a cylinder\( = {\rm{circumference}}\,{\rm{of}}\,{\rm{the}}\,{\rm{curved}}\,{\rm{surface}}\,{\rm{(circle)}} \times {\rm{height}}\,{\rm{of}}\,{\rm{the}}\,{\rm{cylinder}}{\rm{.}}\)

Curved surface area \( = 2\pi rh \ldots \ldots {\rm{(ii)}}\)

The total surface area with radius \(r,\) and height \(h\) is equal to the sum of the curved area and circular areas of the cylinder.

\({\rm{Total}}\,{\rm{surface}}\,{\rm{area}}\,\left( {{\rm{TSA}}} \right) = {\rm{Curved}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{cylinder}} + {\rm{Area}}\,{\rm{of}}\,{\rm{two}}\,{\rm{circles}}…\left( {iii} \right)\)

Substitute the equation \({\rm{(i)}}\) and \({\rm{(ii)}}\) in the equation \({\rm{(iii)}}\)

\({\rm{Total}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{(TSA)}} = 2\pi rh + 2\pi {r^2} \ldots {\rm{(iv)}}\)

Take out the common factors in the equation \({\rm{(iv)}}\)

\({\rm{Total}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{(TSA)}} = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)

Therefore, the surface area formulas are proved.

Volume of a Cylinder

A cylinder is a solid that has three dimensions. In general, a three-dimensional shape’s volume is a function of its base area and height. The product of the area of the circular base and the height of the cylinder equals the volume of the cylinder.

Cubic units are used to measure the volume of a cylinder.

The volume of a cylinder\( = {\rm{Area}}\,{\rm{of}}\,{\rm{circle \times Height}}\)
Area of circle \( = \pi {r^2}\)
The height of the right circular cylinder is \((h).\)
The volume of a cylinder\( = \pi {r^2}h.\)

Solved Examples

Q.1. If there is a cylinder of height \(6\;{\rm{cm}}\) and radius of \(5\;{\rm{cm}}.\) Calculate its total surface area.
Ans: Given: \(h = 6\;{\rm{cm}},r = 5\;{\rm{cm}}\)
The total surface area of a right circular cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
\({\rm{TSA}} = 2 \times 3.14 \times 5(6 + 5)\)
\({\rm{TSA}} = 6.28 \times 5(11)\)
\({\rm{TSA}} = 6.28 \times 55\)
\({\rm{TSA}} = 345.4\,{\rm{sq}}.{\rm{cm}}\)
Hence, the total surface area of a cylinder is \(345.4\,{\rm{sq}}.{\rm{cm}}.\)

Q.2. A cylinder has a height \(5\;{\rm{cm}}\) and a radius \(3\;{\rm{cm}}.\) Find its surface area.
Ans: Given: \({\rm{Radius}}\,r = 3\;{\rm{cm}},{\rm{height}}\,h = 5\;{\rm{cm}}\)
Curved Surface Area of a Cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow {\rm{CSA}} = 2 \times 3.14 \times 3 \times 5\)
\( \Rightarrow {\rm{CSA}} = 94.2\;{\rm{c}}{{\rm{m}}^2}\)
Then, the total surface area of a cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow ({\rm{TSA}}) = 2 \times 3.14 \times 3(5 + 3)\)
\( \Rightarrow ({\rm{TSA}}) = 2 \times 3.14 \times 3(8)\)
\( \Rightarrow {\rm{(TSA)}} = 6.28 \times 24\)
\( \Rightarrow ({\rm{TSA}}) = 150.72\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the curved surface area of the cylinder is \(94.2\;{\rm{c}}{{\rm{m}}^2}\) and total surface area of the cylinder is \(150.72\;{\rm{c}}{{\rm{m}}^2}.\)

Q.3. Anu has given a cylinder of surface area \(1824\pi \) square units. Find the height of the cylinder if the radius of the base of the circle is \({\rm{24}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given: \({\rm{TSA}} = 1824\pi ,r = 24\;{\rm{cm}}\)
Total Surface Area of a Cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow 1824\pi = 2 \times \pi \times 24(h + 24)\)
\( \Rightarrow 1824\pi = 2 \times \pi \times 24(h + 24)\)
\( \Rightarrow 1824\pi = 48\pi (h + 24)\)
\( \Rightarrow \frac{{1824}}{{48}} = (h + 24)\)
\( \Rightarrow 38 = h + 24\)
\( \Rightarrow h = 38 – 24\)
\( \Rightarrow h = 14\)
Hence, the obtained height of the cylinder is \({\rm{14}}\,{\rm{cm}}{\rm{.}}\)

Q.4. The radius of a cylinder is \(4\;{\rm{m}}.\) Find the curved surface area of the cylinder if the height of the cylinder is \(15\;{\rm{m}}.\)
Ans: Given: \({\rm{Radius}}\,r = 4\;{\rm{m}},{\rm{height}}\,h = 15\;{\rm{m}}\)
Curved Surface Area of a Cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow {\rm{CSA}} = 2 \times 3.14 \times 4 \times 15\)
\( \Rightarrow {\rm{CSA}} = 376.8\;{{\rm{m}}^2}\)
Hence, the curved surface area of the cylinder is \(376.8\;{{\rm{m}}^2}\)

Q.5. Find the curved surface area and the total surface area of a cylinder with a base radius of \(4\) inches and a height of \(7\) inches.
Ans: Given: \({\rm{Radius}}\,{\rm{r = 4}}\,{\rm{inches, height}}\,{\rm{h = 7}}\,{\rm{inches}}\)
Curved surface area of a cylinder \(({\rm{CSA}}) = 2\pi {\rm{rh}}\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow {\rm{CSA}} = 2 \times 3.14 \times 4 \times 7\)
\( \Rightarrow {\rm{CSA}} = 175.84\,{\rm{i}}{{\rm{n}}^2}\)
Then, total surface area of a cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
\( \Rightarrow {\rm{(TSA)}} = 2 \times 3.14 \times 4(7 + 4)\)
\( \Rightarrow {\rm{(TSA)}} = 2 \times 3.14 \times 4(11)\)
\( \Rightarrow {\rm{(TSA)}} = 6.28 \times 44\)
\( \Rightarrow {\rm{(TSA)}} = 276.32\,{\rm{i}}{{\rm{n}}^{\rm{2}}}\)
Hence, the curved surface area of the cylinder is \(175.84\,{\rm{i}}{{\rm{n}}^2}\) and total surface area of the cylinder is \(276.32\,{\rm{i}}{{\rm{n}}^{\rm{2}}}.\)

Summary

The entire space covered by the flat surfaces of the cylinder’s bases and the curved surface is known as the surface area of a cylinder. The curved surface area is calculated as the area of the curved face of the cylinder. The total surface area of a cylinder is equal to the sum of areas of all its faces.

This article includes the formulas of the curved surface area and the total surface area of a cylinder and its derivation. The detailed explanation in the article helps to understand the topic better.

Learn the Important Surface Area Formulas

Frequently Asked Questions

We have provided some frequently asked questions here:

Q.1. Who discovered the formula to calculate the volume and surface area of a cylinder?
Ans: Archimedes discovered the formula to calculate the volume and surface area of a cylinder.

Q.2. How do you find the surface area of a cylinder?
Ans: The surface area of a cylinder can be calculated by using the following steps:
Take note of the cylinder’s base radius, \(r,\) and height, \(h.\) Make sure the measurement units are the same.
1. To calculate the surface area of a cylinder, use the formulas:
2. Total surface area of a cylinder \({\rm{(TSA)}} = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
The curved surface area of a cylinder \(\left( {{\rm{CSA}}} \right) = 2\,\pi rh\,{\rm{sq}}{\rm{.units}}\)
3. Express the obtained answer with a suitable unit.

Q.3. What are the three dimensions of a cylinder?
Ans: A cylinder is a three-dimensional shape with two round faces at the top and bottom, as well as one curving surface. A cylinder has a radius and a height.

Q.4. What is the formula for the surface area of a cylinder?
Ans: The surface area of a cylinder formula is,
Total surface area of a cylinder \({\rm{(TSA)}} = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)
The curved surface area of a cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)

Q.5. What is the curved surface area of a cylinder formula?
Ans: Curved surface area of a cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)

Q.6. What is the total surface area of a cylinder formula?
Ans: Total surface area of a cylinder \({\rm{(TSA)}} = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)

We hope this detailed article on the surface area of a cylinder helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!