Volume is the total amount of three-dimensional space that a solid object occupies or fills up.
Unlike area, which only measures a flat two-dimensional surface space, volume adds a third direction: depth. If area measures the flat floor space of a room, volume measures the entire space inside the room from floor to ceiling.
The official SI unit for measuring volume is the cubic meter (m3).
Because volume measures a three-dimensional solid space instead of a flat path, it is always expressed in cubic units.
Other units include cubic millimeters (mm3), cubic centimeters (cm3), cubic decimeters (dm3), and cubic meters (m3).
You must never use square units like cm2 or linear units like cm for volume, because square units are strictly reserved for flat surfaces and linear units are for straight boundaries.
While volume and capacity are closely related, they have a clear difference in practical measurement:
If a solid brick is dropped into water, the amount of space it takes up and pushes aside is its volume (e.g., 1,000 cm3).
If an empty plastic bucket can be completely filled with liquid up to its brim, the total amount of fluid it holds is its capacity (e.g., 1 liter).
Examples of Volume (Space Occupied)
(i) A concrete block occupies 0.5 m3 of construction space.
(ii) A wooden storage crate takes up 2.4 m3 of warehouse space.
(iii) A small decorative brick has a volume of 1,200 cm3.
Examples of Capacity (Container Holding Power)
(i) A carton can hold up to 1,000 mL of milk.
(ii) A large residential storage tank holds 5,000 liters of fresh water.
(iii) A standard fuel tank has a capacity of 55 liters of gasoline.
| Unit Name | Symbol | Value in m3 | Value in Liters (L) |
|---|---|---|---|
| Cubic Millimeter | mm3 | 10-9 m3 | 0.000001 L |
| Cubic Centimeter | cm3 / mL | 10-6 m3 | 0.001 L |
| Cubic Decimeter | dm3 / L | 10-3 m3 | 1 L |
| Cubic Meter | m3 / kL | 1 m3 | 1,000 L |
Solved Example
A plastic storage container holds 2.5 m3 of water. Determine the volume of water inside the container in cubic centimeters.
Solution
The volume value is originally given in m3.
1 m3 = 1,000,000 cm3.
What about 2.5 m3?
We multiply 2.5 m3 by 1,000,000 cm3
1,000,000 cm3 × 2.5 m3 1 m3 = 2,500,000 cubic centimeters
Volume = 2500000 cm3
A regular three-dimensional solid is a shape whose outer surfaces are completely uniform and flat or perfectly curved. To find the space inside these geometric structures, we use specialized volume formulas.
A cube is a three-dimensional solid with six identical square faces. Because all side lengths are exactly equal, its volume is calculated by multiplying its side length by itself three times.
Formula:
For a cube with side length s:
Volume = s3
Where:
Example: Find the volume of a solid cube tracking a uniform edge side length of 4 cm.
Formula:
Volume = s3
s = 4 cm
Calculation:
Volume = 4 × 4 × 4
Volume = 16 × 4
Volume = 64 cm3
A cuboid is a box-like solid shape with six rectangular faces. Its three-dimensional space capacity is found by multiplying its linear length baseline by its horizontal width depth, and then by its vertical height line.
Formula:
For a cuboid with length l, width w, and height h:
Volume = l × w × h
Where:
Example: Find the volume inside a rectangular storage cuboid with a length of 7 m, a width depth of 2 m, and a standing vertical height of 3 m.
Formula:
Volume = l × w × h
l = 7 m
w = 2 m
h = 3 m
Calculation:
Volume = 7 × 2 × 3
Volume = 14 × 3
Volume = 42 m3
A cylinder is a solid with two identical circular bases and a curved side wrapper. To find its volume, multiply the flat area of the circular base by the vertical straight height.
Formula:
For a cylinder with radius r and vertical height h:
Volume = πr2h
Where:
Example: Find the volume of a cylindrical container with a base radius of 7 cm and a vertical height of 10 cm.
Formula:
Volume = πr2h
r = 7 cm
h = 10 cm
Pi (π) = 22/7
Calculation:
Volume = 22 7 × 7 × 7 × 10
Volume = 22 × 7 × 10
Volume = 154 × 10
Volume = 1,540 cm3
A triangular prism has two identical triangular faces held parallel to each other by three rectangular walls. Its volume is calculated by multiplying the area of the triangle ( Cross-sectional area ) by the horizontal length depth of the prism body.
Formula:
For a triangular prism with front triangle base b, triangle height h, and prism depth length L:
Volume = Cross-sectional Area × Length = 1 2 × b × h × L
Where:
Example: Find the volume inside a triangular prism building beam with a horizontal triangle base line of 6 m, a vertical triangle height of 4 m, and a total framework tracking depth length of 12 m.
Formula:
Volume = 1 2 × b × h × L
b = 6 m
h = 4 m
L = 12 m
Calculation:
Volume = 1 2 × 6 × 4 × 12
Volume = 3 × 4 × 12
Volume = 12 × 12
Volume = 144 m3
A pyramid is a 3D solid with a polygonal base and triangular faces that meet at a single point called the apex. Its volume is exactly one-third of the space of a prism with the same base and height.
Formula:
For a pyramid with perpendicular height h:
Volume = 1 3 × Base Area × h
Where:
Example: Find the volume of a square-based pyramid where the bottom base has a side length of 6 cm, and the perpendicular height running up to the apex is 9 cm.
Formula:
Volume = 1 3 × Base Area × h
Base Shape = Square (6 cm × 6 cm)
Base Area = 6 × 6 = 36 cm2
h = 9 cm
Calculation:
Volume = 1 3 × 36 × 9
Volume = 12 × 9
Volume = 108 cm3
A cone is a distinct solid shape featuring a flat circular baseline deck tapering smoothly up to a single pointed apex. Its internal physical capacity is found by multiplying one-third of the circular base surface area by its perpendicular height.
Formula:
For a cone with base radius r and perpendicular height h:
Volume = 1 3 πr2h
Where:
Example: Find the internal volume of a conical storage tent featuring a base radius of 7 m and a perpendicular height of 12 m.( Take π = 22/7 )
Formula:
Volume = 1 3 πr2h
r = 7 m
h = 12 m
Pi (π) = 22/7
Calculation:
Volume = 1 3 × 22 7 × 7 × 7 × 12
Volume = 1 × 22 × 7 × 4
Volume = 154 × 4
Volume = 616 m3
A sphere is a perfectly round 3D geometric solid where every point on the surface is an equal distance from the center. Its internal capacity is calculated by multiplying four-thirds of Pi by the radius cubed.
Formula:
For a sphere with radius r:
Volume = 4 3 πr3
Where:
Example: Find the volume of a solid spherical ball featuring a radius of 3 cm. (Take ( pi = 3.142))
Formula:
Volume = 4 3 πr3
r = 3 cm
r3 = 3 × 3 × 3 = 27
Calculation:
Volume = 4 3 × 3.142 × 27
Volume = 4 × 3.142 × 9
Volume = 36 × 3.142
Volume = 113.11 cm3
A hemisphere is exactly half of a complete sphere, formed by cutting a sphere clean through its center. Its volumetric value is exactly half of a full sphere formula, matching two-thirds of Pi by the radius cubed.
Formula:
For a hemisphere with radius r:
Volume = 2 3 πr3
Where:
Example: Find the internal volume of a hemispherical storage bowl featuring an upper radius of 7 m.( Take π = 22/7 )
Formula:
Volume = 2 3 πr3
r = 7 m
Pi (π) = 22/7
Calculation:
Volume = 2 3 × 22 7 × 7 × 7 × 7
Volume = 2 × 22 × 49 ÷ 3
Volume = 2156 ÷ 3
Volume = 718.67 m3
Real-world geometric design often involves combining multiple basic shapes, calculating empty spaces, or working with scaling changes when models grow or shrink.
A composite solid is an object built by joining two or more basic three-dimensional shapes together. To find the total volume, calculate the volume of each individual part separately and add them together.
Total Volume = Volume of Shape 1 + Volume of Shape 2
A solid toy consists of a cylinder with a base radius of 5 cm and a height of 7 cm, topped by a solid cone of the same radius with a height of 6 cm. Find the total volume.( Take π = 22/7 )
Step 1: Calculate the Cylinder Volume.
Vcylinder = πr2h
Vcylinder = 227 × 5 × 5 × 7
Vcylinder = 22 × 25 = 550 cm3
Step 2: Calculate the Cone Volume.
Vcone = 13 × πr2h
Vcone = 13 × 227 × 5 × 5 × 6
Vcone = 22 × 25 × 0.2857 = 157.14 cm3
Step 3: Sum the individual values together.
Total Volume = 550 + 157.14 = 707.14 cm3
Hollow objects, like metal pipes or empty tubes, contain an outer solid layer shell wrapped around an open center space. To find the volume of the solid material wall itself, subtract the inner empty core volume from the total outer volume.
Volume = π(R2 − r2)h
Where R is the outer radius, r is the inner radius, and h is the perpendicular height length.
Example: A hollow iron cylindrical pipe is 10 meters long. Its outer radius is 6 cm and its inner core radius is 4 cm. Find the volume of iron used to manufacture the pipe.( Take π = 3.142 )
Note: Align units first! Height (h) = 10 m = 1,000 cm.
Solution :
Step 1: Write volume formula of a hallow object.
Volume = π(R2 − r2)h
Step 2: Identify the given values.
π = 3.142
R = 6
r = 4
h = 1000
Step 3: Substitute the values into the formula to get the volume.
Volume = 3.142 × (62 − 42) × 1,000
Volume = 3.142 × (36 − 16) × 1,000
Volume = 3.142 × 20 × 1,000 = 62,840 cm3
An irregular solid object does not have uniform straight edges or predictable curves (like a rough stone). Its volume cannot be solved with regular formulas. Instead, we use the water displacement method to find its volume.
Object Volume = Final Level (V2) − Initial Level (V1)
A student fills a graduated cylinder with water up to the 50 mL mark. After dropping a rough stone completely inside, the water level rises to 85 mL. Determine the exact volume of the stone solid.
Calculation:
Volume of Stone = V2 − V1
Volume of Stone = 85 mL − 50 mL = 35 mL
Since 1 mL = 1 cm3, the final solid volume is 35 cm3
When two geometric solid shapes are completely identical in form but differ in size, they are called mathematically similar bodies. The ratio of their volumes is equal to the cube of their linear scale factor (k).
Where :
V1 = volume of the first solid
V2 = volume of the second solid
s1 = corresponding linear dimension (height, radius, side) of the first solid
s2 = corresponding linear dimension (height, radius, side) of the second solid
k = linear scale factor between the two bodies
Two similar manufacturing metal cones have heights of 4 cm and 8 cm respectively. If the smaller cone has an internal volume of 15 cm3, find the volume of the larger model.
Step 1: Determine the length scale factor (k).
k = slarge ssmall = 8 4 = 2
Step 2: Cube the scale factor to discover the volume multiplier.
k3 = 23 = 8
Step 3: Compute the expanded volume.
Vlarge = Vsmall × k3
Vlarge = 15 × 8
Vlarge = 120 cm3
Example 1 (Rectangular Shipping Container)
A heavy freight shipping container features a single straight length baseline of 12 meters, a horizontal depth width of 3 meters, and a standing vertical ceiling height of 4 meters. Calculate the total volumetric solid space inside the container box.
Solution:
Step 1: Write the volume formula of a cuboid.
Volume = l × w × h
Length (l) = 12 m, Width (w) = 3 m, Height (h) = 4 m
Step 2: Substitute the values into the formula to get the volume.
Volume = 12 × 3 × 4
Volume = 36 × 4
Volume = 144 m3
Final Answer = 144 m3
Example 2 (Industrial Fuel Cylinder Tank)
An industrial fuel storage cylinder features an explicit base radius tracking measurement of 14 centimeters and a standing vertical wall length of 50 centimeters. Calculate the complete structural space capacity inside this liquid container.( Take π = 22/7 )
Solution:
Step 1: Write the volume formula of a cylinder.
Volume = πr2h
Radius (r) = 14 cm, Height (h) = 50 cm
Step 2: Substitute the values into the formula to get the volume.
Volume = 22 7 × 14 × 14 × 50
Volume = 22 × 2 × 14 × 50
Volume = 44 × 700
Volume = 30,800 cm3
Final Answer = 30,800 cm3
Example 3 (Conical Sand Mound Pile)
A perfect cone-shaped pile of gravel is placed on a construction yard floor. It has a base radius of 21 decimeters and a perpendicular height of 10 decimeters. Find its volume. (Take π = 22/7)
Solution:
Step 1: Write the volume formula of a cone.
Volume = 1 3 × πr2hRadius (r) = 21 dm, Height (h) = 10 dm
Step 2: Substitute the values into the formula to get the volume.
Volume = 1 3 × 22 7 × 21 × 21 × 10
Volume = 1 × 22 × 21 × 10
Volume = 22 × 210
Volume = 4,620 dm3
Final Answer = 4,620 dm3
Example 4 (Spherical Water Globe Balloon)
A rigid spherical glass globe designed to contain decorative water displays holds a precise internal radius dimension line of 6 centimeters from its center point out to its shell interface. Calculate the total three-dimensional volume capacity inside the globe. (Take π = 3.142)
Solution:
Step 1: Write the volume formula of a sphere.
Volume = 4 3 × πr3
Radius (r) = 6 cm, Pi (π) = 3.142
Step 2: Substitute the values into the formula to get the volume.
r3 = 6 × 6 × 6 = 216
Volume = 4 3 × 3.142 × 216
Volume = 4 × 3.142 × 72
Volume = 288 × 3.142
Volume = 904.90 cm3
Final Answer = 904.90 cm3
Example 5 (Solid Triangular Prism Metal Wedge)
An industrial mechanical support wedge is manufactured out of solid cast iron. The front cross-section face is a triangle featuring a horizontal baseline profile length of 8 centimeters and a vertical height of 5 centimeters. The horizontal tracking extrusion length depth of the prism body measures exactly 15 centimeters. Calculate the total solid volumetric space inside the wedge block.
Solution:
Base (b) = 8 cm, Height (h) = 5 cm, Prism Length (L) = 15 cm
Step 1: Write the volume formula of a triangular prism.
Volume = Cross-sectional Area × Length
Step 2: Calculate the Cross-sectional area ( Area of the triangle ).
Area = 1 2 × b × hArea = 1 2 × 8 × 5 = 4 × 5 = 20 cm2
Step 3: Multiply the Cross-sectional area by the length of the prism.
Cross-sectional area = 20 cm2
Length = 15 cm
Volume = 20 × 15
Volume = 300 cm3
Final Answer = 300 cm3
Attempt the following questions to test your understanding of volume concepts:
(1.) A solid metal cube has a single side edge measurement of exactly 5 centimeters. Calculate the total three-dimensional volume space occupied by the cube.
(2.) A closed cylinder container features a flat circular base radius of 7 meters and a standing vertical framework height of 8 meters. Find the total volume inside the container. ( Take π = 22/7 )
(3.) A conical funnel tracking system has a baseline radius disk measuring exactly 3 centimeters and stands at an internal perpendicular height of 14 centimeters. Determine its total volumetric capacity. ( Take π = 22/7 )
(4.) A solid rubber playground ball is modeled as a perfect sphere featuring a radius measurement line of 3 meters. Calculate the entire three-dimensional space enclosed inside the ball. ( Take π = 3.142 )
(5.) A water tank is shaped like a cuboid. Its flat horizontal baseline features a length of 6 meters and a depth width of 4 meters. When filled with liquid up to its brim, it stands at a height of 2 meters. Find the total liquid capacity volume inside the tank container.
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