How to find the area of a cylinder base

Now, after we know how to find the surface area of a cylinder, let's derive appropriate formulas for the surface area of a right circular cylinder. To calculate the base surface area, you need to compute the area of a circle with the radius r. But remember that every cylinder has two bases! Thus, you need to multiply it by two:

Table of Contents Show

Circular Cylinder Shape
Calculator Use
Cylinder Formulas in terms of r and h:
Cylinder Calculations:
Learning Outcomes
Volume and Surface Area of a Cylinder

base_area = 2 * π * r²

Estimation of the lateral surface area is even easier. Because the area of a rectangle is the product of its sides, we can write that:

lateral_area = (2 * π * r) * h,

where

2 * π * r is the circumference of the base circle,
h is the height of a cylinder.

Finally, the total surface area of the cylinder formula is simply the sum of the base surface area and the lateral surface area:

total_area = base_area + lateral_area,

or total_area = 2 * π * r² + (2 * π * r) * h,

or total_area = 2 * π * r * (r + h).

With our surface area of a cylinder calculator, you can perform all the calculations in many different units. If you want to learn more about area unit conversion, check out our area converter now!

In the advanced mode of this calculator, you can also calculate the volume of a cylinder, but we also have a dedicated tool called cylinder volume calculator.

💡 The interesting fact is that every cylinder with the same heights and base areas has the same volume. It doesn't matter whether it is a right or oblique cylinder.

Circular Cylinder Shape

r = radius
h = height
V = volume
L = lateral surface area
T = top surface area
B = base surface area
A = total surface area
π = pi = 3.1415926535898
√ = square root

Calculator Use

This online calculator will calculate the various properties of a cylinder given 2 known values. It will also calculate those properties in terms of PI π. This is a right circular cylinder where the top and bottom surfaces are parallel but it is commonly referred to as a "cylinder."

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know r and h in mm, your calculations will result with V in mm3, L in mm2, T in mm2, B in mm2 and A in mm2.

Below are the standard formulas for a cylinder. Calculations are based on algebraic manipulation of these standard formulas.

Cylinder Formulas in terms of r and h:

Calculate volume of a cylinder:
- V = πr2h
Calculate the lateral surface area of a cylinder (just the curved outside)**:
- L = 2πrh
Calculate the top and bottom surface area of a cylinder (2 circles):
- T = B = πr2
Total surface area of a closed cylinder is:
- A = L + T + B = 2πrh + 2(πr2) = 2πr(h+r)

** The area calculated is only the lateral surface of the outer cylinder wall. To calculate the total surface area you will need to also calculate the area of the top and bottom. You can do this using the circle calculator.

Cylinder Calculations:

Use the following additional formulas along with the formulas above.

Given radius and height calculate the volume, lateral surface area and total surface area.
Calculate V, L, A | Given r, h
- use the formulas above
Given radius and volume calculate the height, lateral surface area and total surface area.
Calculate h, L, A | Given r, V
- h = V / πr2
Given radius and lateral surface area calculate the height, volume and total surface area.
Calculate h, V, A | Given r, L
- h = L/2πr
Given height and lateral surface area calculate the radius, volume and total surface area.
Calculate r, V, A | Given h, L
- r = L/2πh
Given height and volume calculate the radius, lateral surface area and total surface area.
Calculate r, L, A | Given h, V
- $r = √( V / πh )

Learning Outcomes

Find the volume and surface area of a cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height [latex]h[/latex] of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[/latex] , will be perpendicular to the bases.

A cylinder has two circular bases of equal size. The height is the distance between the bases.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, [latex]V=Bh[/latex] , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, [latex]B[/latex] , is the area of the rectangular base, length × width. For a cylinder, the area of the base, [latex]B[/latex], is the area of its circular base, [latex]\pi {r}^{2}[/latex]. The image below compares how the formula [latex]V=Bh[/latex] is used for rectangular solids and cylinders.

Seeing how a cylinder is similar to a rectangular solid may make it easier to understand the formula for the volume of a cylinder.

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See the image below.

By cutting and unrolling the label of a can of vegetables, we can see that the surface of a cylinder is a rectangle. The length of the rectangle is the circumference of the cylinder’s base, and the width is the height of the cylinder.

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length [latex]L[/latex] of the rectangular label. The height of the cylinder is the width [latex]W[/latex] of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius [latex]r[/latex] and height [latex]h[/latex], is

[latex]S=2\pi {r}^{2}+2\pi rh[/latex]

Volume and Surface Area of a Cylinder

For a cylinder with radius [latex]r[/latex] and height [latex]h:[/latex]

example

A cylinder has height [latex]5[/latex] centimeters and radius [latex]3[/latex] centimeters. Find the 1. volume and 2. surface area.

Solution

Step 1. Read the problem. Draw the figure and label

it with the given information.

1.
Step 2. Identify what you are looking for.	the volume of the cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] )	[latex]V=\pi {r}^{2}h[/latex] [latex]V\approx \left(3.14\right){3}^{2}\cdot 5[/latex]
Step 5. Solve.	[latex]V\approx 141.3[/latex]
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately [latex]141.3[/latex] cubic inches.

2.
Step 2. Identify what you are looking for.	the surface area of the cylinder
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] )	[latex]S=2\pi {r}^{2}+2\pi rh[/latex] [latex]S\approx 2\left(3.14\right){3}^{2}+2\left(3.14\right)\left(3\right)5[/latex]
Step 5. Solve.	[latex]S\approx 150.72[/latex]
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately [latex]150.72[/latex] square inches.

try it

example

Find the 1. volume and 2. surface area of a can of soda. The radius of the base is [latex]4[/latex] centimeters and the height is [latex]13[/latex] centimeters. Assume the can is shaped exactly like a cylinder.