Surface Area of a Cylinder Calculator | Free Online Surface Area of a Cylinder Calculator with Steps - BYJUS

# Surface Area of a Cylinder Calculator

The surface area of a cylinder calculator is a free online tool that helps us calculate the surface area of a cylinder as well as its radius, height or its lateral surface area. Let us familiarize ourselves with the calculator....Read MoreRead Less

## Online Surface Area of a Cylinder Calculator

### How to Use the ‘Surface Area of a Cylinder Calculator’?

Follow the steps below to use the surface area of a cylinder calculator:

Step 1: Enter the known measure or the measures into the respective input box or in multiple boxes, and the unknown measure or measures will be calculated.

Step 2: Select the appropriate units for the input and output.

Step 3: You can also select the desired value of ‘pi’ from the dropdown box. The values can be the following: 3.14, $$\pi$$  or $$\frac{22}{7}$$.

Step 4: Click on the ‘Solve’ button to obtain the result.

Step 5: Click on the ‘Show steps’ button to know the stepwise solution to find the missing measures.

Step 6: Click on the button to enter new inputs and start again.

Step 7: Click on the ‘Example’ button to play with different and random input values.

Step 8: Click on the ‘Explore’ button to visualize the splitting of a cylinder into a rectangle and two circles, and thereby calculate the surface area as well as the lateral surface area.

Step 9: When on the ‘Explore’ page, click  the ‘Calculate’ button if you want to go back to the calculator.

### What is a Cylinder?

A cylinder is a three dimensional figure consisting of two congruent circular bases that are joined together by a curved surface. The perpendicular distance between the circular faces is called the height of the cylinder. The curved surface is called the lateral surface of the cylinder.  ### Formulas used for the ‘Surface Area of a Cylinder Calculator'

Consider a cylinder of radius r and height h. Lateral Surface Area of cylinder, LSA = 2$$\pi$$rh

• When lateral surface area LSA and height h is known, the radius of cylinder, r = $$\frac{LSA}{2\pi h}$$
• When lateral surface area LSA and radius r is known, the height of cylinder, h = $$\frac{LSA}{2\pi r}$$

Surface Area of a cylinder, SA = LSA + $$2\pi r^2$$

• When surface area SA and height h is known, the radius r of the cylinder can be calculated in the following manner:

SA = LSA + $$2\pi r^2$$

SA = $$2\pi rh$$ + $$2\pi r^2$$

$$2\pi r^2$$ + $$2\pi rh$$ – SA = 0        (1)

Therefore, this is a quadratic equation, where r is the variable.

A quadratic equation a$$x^2$$ + b$$x$$ + c = 0, can be solved for x as:

$$x=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$$

On applying the above expression to equation (1):

2$$\pi r^2$$ + 2$$\pi$$rh – SA = 0

a = 2$$\pi$$, b = 2$$\pi$$h and c = – SA

Radius is a measure of length, hence, the radius will always be positive. Therefore, we consider only the positive root of the equation.

Radius of cylinder, $$x=\frac{-2\pi h+\sqrt{(2\pi h)^2+8\pi(SA)}}{4\pi}$$

• When surface area SA and lateral surface area LSA is known, the radius of cylinder, $$r=\sqrt{\frac{SA-LA}{2\pi}}$$
• When surface area SA and radius r is known, the height of cylinder, $$h=\frac{SA-2\pi r^2}{2\pi r}$$

### Derivation of the lateral surface area and surface area of cylinder formula

Consider a cylinder of radius r and height h. If we unravel the cylinder or vertically cut through, we get two circles and a curved surface in the shape of a rectangle. The length of the rectangle will be equal to the circumference of the base, that is, 2$$\pi$$r and its width will be the height of the cylinder, h.  Lateral Surface Area of a cylinder is the area of its lateral surface.

Therefore, the area of the rectangle will be equal to the lateral surface area of the cylinder.

Area of rectangle = length $$\times$$ width

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

= 2$$\pi$$rh

= LSA

Surface Area of a cylinder is the area of its lateral surface added to the area of its bases.

Therefore, the surface area, SA = LSA + area of two circles

= $$2\pi rh+2\times \pi r^2$$

= $$2\pi rh+2\pi r^2$$

### Solved Examples

Example 1: Find the surface area and the lateral surface area of a cylinder, if its radius is 14 inches and height is 17 inches. (Take $$\pi=\frac{22}{7}$$)

Solution:

Stated in the question:

r = 14 inches

h = 17 inches

Lateral Surface area, LSA = 2$$\pi$$rh

= $$2\times\frac{22}{7}\times 14\times 17$$

= 1496 square inches

We know that, Surface area $$SA=LSA+2\pi r^2$$

$$SA=1496+2\times \frac{22}{7}\times 14^2$$

$$SA=2728$$ square inches

So, the lateral surface area of the cylinder is 1496 square inches and the surface area of the cylinder is 2728 square inches.

Example 2: Find the radius and surface area of a cylindrical drum, if its lateral surface area is 1320 square millimeters and height is 21 millimeters.

Solution:

Details in the question:

LSA = 1320 $$mm^2$$

h = 21 mm

LSA = 2$$\pi$$rh

Rearrange the formula to find the radius as in,

r = $$\frac{LSA}{2\pi h}$$

r = $$\frac{1320}{2\times \frac{22}{7} \times 21}$$

r = 10 mm

Surface area, SA = LSA + 2$$\pi r^2$$

= 1320 + 2$$\times \frac{22}{7}\times 10^2$$

= 1948.571 $$mm^2$$

So, the radius of the drum is 10 millimeters and its surface area is 1948.571 square millimeters.

Example 3: Find the height and surface area of a cylindrical water bottle, whose lateral surface area is 88 square inches and radius is 1.4 inches.

Solution:

As given in the question:

LSA = 88 $$in ^2$$

r = 1.4 in

Rearrange the formula of lateral surface area for height as in,

LSA = 2$$\pi$$rh

h = $$\frac{LSA}{2\pi r}$$

h = $$\frac{88}{2\times \frac{22}{7}\times 1.4}$$

h = 10 inches

Surface area, SA = LSA + 2$$\pi r^2$$

= 88 + $$2 \times \frac{22}{7}\times 1.4^2$$

= 100.32 $$in^2$$

So, the height of the water bottle is 10 inches and its surface area is 100.32 square inches.

Example 4: Find the height and lateral surface area of a cylinder that has a radius of 5 inches and a surface area of 250 square inches.

Solution:

The details provided are:

r  = 5 in

SA = 250 $$in^2$$

SA = 2$$\pi$$rh+2$$\pi r^2$$

Rearrange the formula of surface area to find the height as in,

h = $$\frac{SA-2\pi r^2}{2\pi r}$$

h = $$\frac{(250-2\times 3.14\times 5^2)}{2\times 3.14 \times 5}$$

h = 2.962 inches

Applying the formula for lateral surface area:

LSA = 2$$\pi$$rh

LSA = 2 $$\times$$ 3.14 $$\times$$ 5 $$\times$$ 2.962

LSA = 93 $$in^2$$

So, the height of the cylinder is 2.962 inches and its lateral surface area is 93 square inches.

Example 5: Find the radius and lateral surface area of a cylinder if its height is 4 yards and surface area is 200 square yards.

Solution:

Given are the following details:

h = 4 yd

SA = 200 $$yd^2$$

SA = 2$$\pi$$rh + 2$$\pi r^2$$

Rearrange the formula of surface area to find the radius as in:

r = $$\frac{-2\pi h+\sqrt{(2\pi h)^2+8\pi(SA)}}{4\pi}$$

r = $$\frac{-2\times 3.14 \times 4+\sqrt{(2\times 3.14 \times 4)^2+8\times 3.14 \times 200}}{4\times 3.14}$$

r = 3.987 yards

Using the formula for lateral surface area:

LSA = 2$$\pi$$rh

LSA = 2$$\times$$ 3.14 $$\times$$ 4 $$\times$$ 3.987

LSA = 100.16 $$yd^2$$

So, the radius and lateral surface area of the cylinder is 3.987 yards and 100.16 square yards respectively.

Example 6: Find the radius and height of a cylinder, if the lateral surface area is 50 square feet and surface area is 80 square feet.

Solution:

Given:

LSA = 50 $$ft^2$$

SA = 80 $$ft^2$$

SA = LSA + 2$$\pi r^2$$

Rearrange the formula of surface area to find the radius as in,

r = $$\sqrt{\frac{SA-LSA}{2\pi}}$$

r = $$\sqrt{\frac{80-50}{2\times 3.14}}$$

r = 2.186 yards

Rearrange the lateral surface area formula to find the height as in,

LSA = 2$$\pi$$rh

h = $$\frac{LSA}{2\pi r}$$

h = $$\frac{50}{2\times 3.14 \times 2.186}$$

h = 3.642 yards

So, the radius of the cylinder is 2.186 yards and height is 3.642 yards.