How To Find The Volume Of A Cylinder
Definition: The number of cubic units that will exactly fill a cylinder
How to find the volume of a cylinder
Although a cylinder is technically not a prism, it shares many of the properties of a prism. Like prisms, the volume is found by multiplying the area of one end of the cylinder (base) by its height.
Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula:
area = π r2
Multiplying by the height h we get
volume = π r2
h
where:
π is Pi, approximately 3.142
r is the radius of the circular end of the cylinder
h height of the cylinder
Volume of a partially filled cylinder
One practical application is where you have horizontal cylindrical tank partly filled with liquid. Using the formula above you can find the volume of the cylinder which gives it’s maximum capacity, but you often need to know the volume of liquid in the tank given the depth of the liquid.
This can be done using the methods described in Volume of a horizontal cylindrical segment
Oblique cylinders
Recall that an oblique cylinder is one that ‘leans over’ - where the top center is not over the base center point. In the figure above check "allow oblique’ and drag the top orange dot sideways to see an oblique cylinder.
It turns out that the volume formula works just the same for these. You must however use the perpendicular height in the formula. This is the vertical line to left in the figure above. To illustrate this, check ‘Freeze height’. As you drag the top of the cylinder left and right, watch the volume calculation and note that the volume never changes.
See Oblique Cylinders for a deeper discussion on why this is so.
Units
Remember that the radius and the height must be in the same units - convert them if necessary. The resulting volume will be in those cubic units. So, for example if the height and radius are both in centimeters, then the volume will be in cubic centimeters.
Things to try
- In the figure above, click ‘reset’ and ‘hide details’
- Drag the two dots to alter the size and shape of the cylinder
- Calculate the volume of that cylinder
- Click ‘show details’ to check your answer.
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 ).
Volume of a Cylinder
A cylinder is a solid composed of two congruent circles in parallel planes, their interiors, and all the line segments parallel to the segment containing the centers of both circles with endpoints on the circular regions.
The volume of a 33 -dimensional solid is the amount of space it occupies. Volume is measured in cubic units ( in3,ft3,cm3,m3in3,ft3,cm3,m3 , et cetera). Be sure that all of the measurements are in the same unit before computing the volume.
The volume VV of a cylinder with radius rr is the area of the base BB times the height hh .
V=Bh or V=πr2hV=Bh or V=πr2h
Example:
Find the volume of the cylinder shown. Round to the neatest cubic centimeter.
Solution
The formula for the volume of a cylinder is V=Bh or V=πr2hV=Bh or V=πr2h .
The radius of the cylinder is 88 cm and the height is 1515 cm.
Substitute 88 for rr and 1515 for hh in the formula V=πr2hV=πr2h .
V=π(8)2(15)V=π(8)2(15)
Simplify.
V=π(64)(15)≈3016V=π(64)(15)≈3016
Therefore, the volume of the cylinder is about 30163016 cubic centimeters.
