# Curved Surface Area of Cylinder Formula & Derivation (2023)

The total surface area or simply the surface area of a cylinder is described as the space covered by the flat surface of the cylinder's top and bottom bases and the curved surface of the cylinder. The total area of the cylinder includes the area of the top and bottom circular bases of the cylinder plus the area of the curved surface. The area and diameter of both circular bases are the same.

There are many examples of cylindrical shaped objects that we use in our daily life like pipes, candles, tanks, wells etc.

## What is a Cylinder?

A cylinder is a three-dimensional object. It has two parallel circular bases at each end of the cylindrical shape. The line connecting the centers of these circular bases is known as the axis.

The distance between the axis and the other half of the outer part is called the radius of the cylinder. The distance between the two circular bases is the height of the cylinder.

## Curved Surface Area of a Cylinder

There are two types of cylinder: a right circular cylinder and oblique cylinder as shown in the image below. Thus we need to learn about the curved surface area of a right circular cylinder and that of an oblique cylinder. However, note that we will refer to the curved surface area of the right circular cylinder simply as the curved surface area of the cylinder. The area of the curved surface of the cylinder is said to be the curved surface area of the cylinder. This can be obtained when we exclude the top and bottom circular areas from the total area of the cylinder.

## Curved Surface Area of Cylinder Formula

So what is the formula of curved surface area of cylinder? The formula for the curved surface area of cylinder is represented as:

The curved surface area of cylinder= $$2\pi m n$$

Where, m = radius of the cylinder

n = height of the cylinder

## Derivation of the Formula for Surface Area of Cylinder

In the above provided figure of a cylinder in which the area of the curved surface opens as a rectangle and the two bases which are at top and bottom are circular bases..

Let a cylinder whose radius is r and height is h.

The cylinder is divided into three parts: top circular base, a rectangle , and bottom circular base.

The rectangle has a length of 2πr and a width of h. Therefore, the area of the rectangle, P1 = $$2\pi r h$$, which is equal to the area of the curved surface of the cylinder.

The area of top circular base with radius r is given by $$\pi r^{2}$$ and the area of bottom circular base with radius r is given by $$\pi r^{2}$$. Therefore, the area of two of top and bottom bases, P2 =$$\pi r^{2}+\pi r^{2}= 2\pi r^{2}$$.

Now the total surface area of the cylinder = P1 + P2

= $$2\pi r^{2}+2\pi r h$$

(Video) Cylinder - Curved Surface Area, Total Surface Area & Volume of Cylinder Proof with Activity

= $$2\pi\times r\times(r+h)$$

Therefore, the total surface area of the cylinder is $$2\pi\times r\times(r+h)$$ .

## How to find Curved Surface Area of Cylinder?

The surface area of a cylinder is the area occupied by the top and bottom bases of the cylinder and the area of the curved surface of the cylinder.

We use the following steps to find the total surface area of a cylinder with a radius of 20 cm and a height of 15 cm.

Step 1: Let the radius "m" and the height "n" of the cylinder. The unit of radius and height should be the same. Here, given m= 20 cm, n= 15 cm

Step 2: We require to find the total surface area of the cylinder. Therefore,we apply the formula for the total surface area of the cylinder = $$2\pi\times m\times(m+n)$$.

Step 3: We put the value of m and n in the above formula to get the answer in square cm. Putting the value of m and n into the above formulas ,we get.

Total surface area =$$2\pi\times m\times(m+n)$$

=$$2\pi\times 20\times(20+15)$$

=$$2\pi\times 20\times 35$$

=$$2\pi\times 700$$

=$$2\3.14\times 700$$

=$$4396 square cm$$

So the total surface area of a cylinder is 4396 square cm.

## Curved surface area of hollow cylinder

A hollow cylinder which is empty from inside has some difference between the internal and external radius.

Assume r1 be the outer radius of the cylinder and r2 be its inner radius and h be its height.

Thus the curved surface area of a hollow cylinder is the sum of the curved surface area of outer cylinder and inner curved surface area of cylinder.

(Video) What is the Surface Area of a Cylinder? | Don't Memorise

The Curved surface area of hollow cylinder formula is represented as :

The curved surface area of the hollow cylinder=$$2\pi \ r_{1}h+2\pi\ r_{2}h$$.

## Summary ofCurved Surface Area of Cylinder

• The surface area of a cylinder is described as the space covered by the flat surface of the cylinder's top and bottom bases and the curved surface of the cylinder
• The area of the curved surface of the cylinder is said to be the curved surface area of the cylinder.
• The curved surface area of the cylinder= $$2\pi m n$$. Where, m = radius of the cylinder.
• and n = height of the cylinder.
• The total surface area of the cylinder is $$2\pi\times r\times(r+h)$$ .Where, r = radius of the cylinder and h= height of the cylinder.
• The curved surface area of the hollow cylinder=$$2\pi \ r_{1}h+2\pi\ r_{2}h$$.Where, r1 be the outer radius of the given cylinder and r2 be its inner radius and h be its height.

## Curved Surface Area of Cylinder Solved Example

Ex.1.The curved surface area of a cylinder is 264 sq.m and its volume is 924 cub.m. find the ratio of its height to its diameter.

A1. Given, the curved surface area of a cylinder = 264 sq.m and the volume of the cylinder = 924 cub.m.

We Know that the curved Surface area of the cylinder $$2\pi\times r\times h$$ and

Volume of the cylinder = $$\pi r^{2} h$$.

So, $$\frac{ab}{cd}=\frac{924}{264}$$.

$$\frac{r}{2}=\frac{7}{2}$$.

r= 7 meter

The curved surface area of a cylinder= $$2\pi\times r\times h$$

$$264= 2\pi\times r\times h$$

$$264= 2\3.14\times 7\times h$$

$$h=\frac{264}{44}$$.

h=6 meter

Therefore,the ratio of its height to its diameter, $$\frac{h}{2r}=\frac{6}{(2\times 7)}=\frac{3}{7}$$.

Thus,the ratio of its height to its diameter is [latex \frac{3}{7} \).

(Video) Volume of a Cylinder and Surface Area of a Cylinder

Ex-2. The curved surface area of a cylinder is 1320 cm^2 and the circumference of its base is 220 cm. Find the height and the volume of the cylinder.

A2. Given,Circumference of its base = 220cm

$$2\pi r=220$$

$$r=\frac{220}{2\pi}$$

=$$\frac{220}{(2\times 3.14)}$$

=$$\frac{220}{6.28}$$

=$$35 cm$$

Given,Curved Surface Area of the cylinder =1320 cm2

$$2\pi\times r\times h=1320$$

$$h=\frac{1320}{2\pi\times35 }$$

=$$\frac{1320}{(70\times 3.14)}$$

=$$\frac{1320}{219.8}$$

=$$6 cm$$

Volume of the cylinder = $$\pi r^{2} h$$

$$\pi r^{2} h=\pi\times 35^{2}\times 6$$

=$$\pi\times 1225\times 6$$

= $$23,079 cm^3$$

(Video) Cylinder volume and surface area | Perimeter, area, and volume | Geometry | Khan Academy

Therefore,the height and the volume of the cylinder are 20 cm and 76930 cm^3 respectively.

Ex- 3: The radius of a cylinder is 10 inches and the height of the cylinder is 15 inches. Find the surface of the cylinder. (Take Pi like 3.14)

A3 :Given,radius, r = 10 inches

cylinder height, h = 15 inches

The area of the cylinder = $$2\pi\times r\times(r+h)$$

=$$2\pi\times 10\times(10+15)$$

=$$2\pi\times 10 \times 25$$

= $$2\ 3.14\times 10 \times 25$$

= $$1570 square inches$$ .

So, the area of the cylinder is 1570 square inches.

Ex- 3: David has a cylinder with an area of 1728π square units. Find the height of the cylinder when the radius of the circle's base is 12 units.

A3 : cylinder surface, A=1728π; radius (r) = 24; h = ?

Putting the values into the formula to find the height of the cylinder.

$$p =2\pi\times r\times(r+h)$$

$$1728 \pi = 2\pi\times 12\times(12+h)$$

$$\frac{1728}{\pi} = ( 12 + h)$$

$$72 = (12+ h)$$

(Video) To derive the formula for total surface area of a cylinder and volume of a cylinder |

$$h = 60 units$$

So the height of the cylinder is 60 units.

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## FAQs

### What is derivation of curved surface formula? ›

We can also find the curved surface area in terms of height h of the frustum using Pythagoras theorem as A=π(r1+r2)√(r1−r2)2−h2.

How do you derive the formula of a cylinder? ›

Volume of a Cylinder Derivation:

Let us say that the circular discs are stacked up to the height of 'h'. Now, the volume of cylinder will be the product of the base area of the discs and the height 'h'. Thus, the volume of a cylinder of height 'h' and base radius 'r' is given as πr2h.

How do you derive the CSA of a cone? ›

If the radius of the base of the cone is "r" and the slant height of the cone is "l", the surface area of a cone is given as: Total Surface Area, T = πr(r + l) square units. Curved Surface Area, S = πrl square units.

What is the formula for curved surface area? ›

What is the formula of curved surface area? The area obtained after excluding the circular area from the total area of the cylinder is known as curved surface area. Curved surface area is given by 2 * pi *r * h.

What is the derivation of refraction for curved surfaces? ›

For refraction at a curved surface, the u-v relationship is: μ2v−μ1u=μ2−μ1R. μ2 is the refractive index of the medium, at which the refraction takes place. μ1 is the refractive index of the medium, from where the light is incident on the surface. R is the radius of curvature of the refractive surface.

How volume of cylinder can be derived from its area? ›

Calculate the area of the base (which is a circle) by using the equation πr² where r is the radius of the circle. Then, multiply the area of the base by the height of the cylinder to find the volume.

How do you derive the volume of a cone without calculus? ›

This is our grand formula for the volume of a cone: 𝜋𝑟 squared ℎ divided by three or a third 𝜋𝑟 squared ℎ plus 𝜋𝑟 squared ℎ over twelve 𝑛 squared.

How do you derive the volume of a cone in calculus? ›

The formula for the volume of a cone is ⅓ 𝜋r2h cubic units, where r is the radius of the circular base and h is the height of the cone.

How do you derive the volume of a truncated cone? ›

V = (1/3) * π * h * (r² + r * R + R²), Where: R is the radius of the base of the original cone (bottom surface);

What is the formula for the curved surface area and volume of a cylinder? ›

A cylinder's volume is π r² h, and its surface area is 2π r h + 2π r².

### What is curved surface formula and explain the terms? ›

The curved surface formula is, μ2v−μ1u=μ2−μ1R. where μ2 is the refractive index of the medium in which the refraction takes place and μ1 is the refractive index of the medium, from where the light is incident on the surface and. R is the radius curvature of the surface.

How do you calculate curved curvature? ›

The curvature(K) of a path is measured using the radius of the curvature of the path at the given point. If y = f(x) is a curve at a particular point, then the formula for curvature is given as K = 1/R.

What is Snell's law derivation? ›

Snell's law can be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light.

How do you derive the refractive index formula? ›

Refractive index is also equal to the velocity of light c of a given wavelength in empty space divided by its velocity v in a substance, or n = c/v.

Is Snell's law valid for curved surfaces? ›

Hence, Snell's law of ray optics is valid for all the points on a curved surface (interface between the two optical media). And yes, the fermat's principle of least time on plane surfaces can now be used easily.

What is curved surface area of a cylinder? ›

Curved surface area of a cylinder = 2πrh.

What is the difference between surface area and curved surface area? ›

Basically, the surface area can be classified as: Curved Surface Area (CSA) – It includes the area of all the curved surfaces. Lateral Surface Area (LSA) – It includes the area of all the surface excluding the top and bottom areas.

How do you find the surface area of a cylinder without height? ›

Remember, a cylinder is a special kind of prism with circular bases, so you can substitute πr2 for B (area of the base) and 2πr for p (perimeter of the base). The formula for surface area of a cylinder is SA=2πr2+2πrh.

What are the two methods for finding the volume of a cylinder? ›

Thus, the volume of the cylinder can be given by the product of the area of base and height. For any cylinder with base radius 'r', and height 'h', the volume will be base times the height. Therefore, the volume of a cylinder = πr2h cubic units.

What is the derivative of volume of a cylinder with respect to height? ›

A cylinder is just a circle extruded into the third dimension through a height. Since volume is area times length, all we need is the area of a circle (pi r2) and the height h. So V = pi r2 h.

### How do you find the external curved surface of a cylinder? ›

The curved surface area of a cylinder is calculated using the formula, curved surface area of cylinder = 2πrh, where 'r' is the radius and 'h' is the height of the cylinder.

Is curved and lateral surface area same? ›

Lateral surface area is also called as curved surface area.

Why is the derivative of the volume of a cube not the surface area? ›

How about a cube with volume V=x^3? In this case the derivative is 3x^2 so it is not the surface area. Why is this? It is due to symmetry i.e. it depends on whether the volume increases symmetrically when you increase your variable such as length of side, radius, or height, etc.

How do you work out the volume of a cylinder without a calculator? ›

Volume of a cylinder
1. V = A h.
2. Since the area of a circle = π r 2 , then the formula for the volume of a cylinder is:
3. V = π r 2 h.

How is the formula for the volume of a cube derived? ›

The volume of a cube can be found by multiplying the edge length three times. For instance, if the length of an edge of a cube is 4, the volume will be 43. The formula to calculate the volume of a cube is given as, Volume of a cube = s3, where 's' is the length of the side of the cube.

What is curved surface area of truncated cone? ›

Answer: The Curved Surface Area (CSA) of the frustum of a cone is: = pi * l(R + r) where the (r) stands for = radius of the smaller circle and (R) stands for = radius of the bigger circle and the (l) = slant height of the frustum.

What is the formula for truncated cylinder? ›

Truncated cylinder is the geometric solid produced when a cylinder is cut by a plane that is not parallel to the base. The volume of a truncated circular cylinder is V=πr2(h1+h2)2, where h1 and h2 are the lengths of the longest and shortest elements of the cylinder and r is the radius of the base.

What is the image formation derivation of curved surface? ›

the image formation formula for curved surface is n2/v=n1/u=n2-n1/R .

How is the radius of curvature formula derived? ›

The Formula for the Radius of Curvature

R= 1/K, where R is the length or radius of curvature and K is the derivative of curvature.

What is the origin of shape of a curved surface of cylinder? ›

In case one of the circular bases is displaced and the axis does not produce the right angle to the base, then it is called the oblique cylinder. In the middle of the two circular bases, there is a curved surface, which, when opened represents a rectangular shape. This curved surface is also called a lateral surface.

### What is image formation formula? ›

We can find the number of images formed if we know the angle between the plane mirrors. This is given by the formula: Number of images = 360θ−1 if 360θ is even. The number of images = 360θ if it comes odd and when the object is placed asymmetrically.

What is the derivative of curvature? ›

Curvature can actually be determined through the use of the second derivative. When the second derivative is a positive number, the curvature of the graph is concave up, or in a u-shape. When the second derivative is a negative number, the curvature of the graph is concave down or in an n-shape.

What is the formula for radius of curvature for curved surface? ›

The curvature is the reciprocal of the radius of curvature of the curve at a given point. The radius of curvature formula is R=(1+(dydx)2)3/2|d2ydx2| R = ( 1 + ( d y d x ) 2 ) 3 / 2 | d 2 y d x 2 | .

What is curvature formula? ›

The curvature, denoted κ, is one divided by the radius of curvature. In formulas, curvature is defined as the magnitude of the derivative of a unit tangent vector function with respect to arc length: κ = ∣ ∣ d T d s ∣ ∣ \kappa = \left|\left| \dfrac{dT}{ds} \right|\right| κ=∣∣∣∣∣∣∣∣∣∣dsdT∣∣∣∣∣∣∣∣∣∣

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