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.
A cylinder has two circular bases and one curved surface.
The surface area of any figure is expressed in units of area that are expressed in square units such as, square millimeters, square centimeters, square feet or square yards.
The bases of a cylinder are in the shape of a circle.
The perpendicular distance between the bases is called the height of the cylinder.