How to Calculate Volume of a Cube Step by Step
The concept of volume of a cube is central in mathematics as it helps us measure the space inside a perfect 3D square, with applications ranging from everyday objects like dice and ice cubes to exam problems in geometry and physics.
What Is Volume of a Cube?
A cube is a three-dimensional solid shape with six equal, square faces. The volume of a cube is the total space enclosed by all its sides—imagine filling a box or ice cube completely with water, that’s its volume. You’ll find this concept applied in geometry, measurement conversions, and even in everyday packaging or storage calculations.
Key Formula for Volume of a Cube
Here’s the standard formula for finding the volume of a cube:
Volume of Cube = a³
where a is the length of one side of the cube (in units like cm, m, or inches).
Cross-Disciplinary Usage
The volume of a cube is useful beyond just mathematics—it also appears in physics (measuring materials), computer science (memory and storage concepts), and logical reasoning. If you are preparing for exams like JEE or NEET, expect to see questions related to cubic measurement, capacity, and comparison with other 3D shapes.
Step-by-Step Illustration
- Suppose the side of a cube is 5 cm.
Substitute into the formula: Volume = 5 × 5 × 5 - Calculate stepwise:
5 × 5 = 25; then 25 × 5 = 125 - Final Answer:
The volume of the cube is 125 cm³
Units and Conversion Table
The result for volume of a cube is always in cubic units. Here are some common units and conversion values:
| Unit | Symbol | Equivalent |
|---|---|---|
| Cubic centimetres | cm³ | 1 cm³ = 1 mL |
| Cubic metre | m³ | 1 m³ = 1000000 cm³ = 1000 L |
| Cubic inches | in³ | 1 in³ ≈ 16.387 cm³ |
| Litres | L | 1 L = 1000 cm³ |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for exams: If a cube’s side is a decimal or fraction, use multiplication tables or fast rounding for cube calculations. For example, if a = 2.5 cm, then a³ = 2.5 × 2.5 × 2.5 = 15.625 cm³, which can be quickly done using calculator or mental math blocks. It helps to break down the steps (first 2.5 × 2.5, then multiply by 2.5).
Tip: For unit conversions, always convert the side length into the desired unit before cubing. Eg: convert cm to m, then calculate volume in m³.
Frequent Errors and Misunderstandings
- Confusing surface area (a² × number of faces) with volume (a³).
- Forgetting to cube the side length—sometimes students multiply by 2 or square only.
- Mistaking between units (like answering in cm instead of cm³).
- Mixing up volume formulas for cube and cuboid.
Volume of Cube vs. Cuboid (Table)
| Shape | Formula | Key Point |
|---|---|---|
| Cube | a³ | All sides equal |
| Cuboid | l × b × h | Length, breadth, height may differ |
Applications & Real-Life Uses
Understanding the volume of a cube is helpful for finding:
- How much water fits in a cubical tank
- Volume of storage boxes, dice, or ice cubes
- Measuring packaging for cubes in shipping & logistics
In board exams and olympiads, students often calculate the number of smaller cubes that fit into a big cube, or vice versa.
Practice: Solve These Cube Volume Problems
- Find the volume of a cube with side 7 cm.
- If a cube contains 64 cm³, what is the length of each side?
- Compare volumes: a cube with a = 4 cm vs a cuboid with 2 cm × 4 cm × 2 cm.
Relation to Other Concepts
Mastery of the volume of a cube makes calculating volumes of other 3D shapes like cuboids and spheres easier. It also helps with surface area understanding and is a base for learning volume formulas of various geometry solids.
Online Tools and More Resources
- Volume of Cuboid Calculator – Easily compare cube vs cuboid volumes.
- Cube Root Calculator – Quickly find side length when volume is known.
- Area of Square – Review base concepts about the cube’s faces.
- Units Converter – Convert between cm³, m³, and litres easily.
We explored volume of a cube: from meaning and formula to worked examples, common mistakes, and real-world uses. For more practice and doubt clearance, attend Vedantu’s live online Maths sessions or use our free calculators to boost your confidence and speed.
FAQs on Volume of a Cube – Formula, Steps & Solved Examples
1. What is the formula for the volume of a cube?
The volume (V) of a cube is calculated using the formula: V = a³, where 'a' represents the length of one side of the cube.
2. How do you calculate the volume of a cube if the side length is given in centimeters?
1. **Measure** the side length (a) of the cube in centimeters.
2. **Cube** the side length: a x a x a = a³
3. The result, expressed in cubic centimeters (cm³), is the volume of the cube.
3. What is the difference between the volume of a cube and a cuboid?
A cube has all sides equal in length (a), so its volume is a³. A cuboid has three different side lengths (length, breadth, height), so its volume is calculated as length x breadth x height. A cube is a special type of cuboid where all sides are equal.
4. How can I quickly calculate the volume of a cube for my exam?
Memorize the formula V = a³. Practice substituting values quickly. If a calculator is allowed, use it efficiently to cube the side length. For mental calculations, learn efficient cubing techniques or use approximation if accuracy isn't critical.
5. Is the volume of a cube always in cubic units?
Yes, the volume of a cube is always expressed in cubic units because it measures three-dimensional space. Common units include cubic centimeters (cm³), cubic meters (m³), cubic inches (in³), etc.
6. How does the volume of a cube relate to its surface area?
The surface area of a cube is 6a² (six squares with side 'a'). The volume is a³. There's no direct simple relationship between them except that both depend on the side length 'a'.
7. Can a cube have a fractional or decimal side length? How is volume calculated in those cases?
Yes, a cube can have a fractional or decimal side length. The volume is calculated the same way: cube the side length (a³). The result will be a fractional or decimal value representing the volume in cubic units.
8. How do unit conversions (like cm³ to liters) affect volume calculations?
You need to use appropriate conversion factors. Remember that 1 liter = 1000 cm³. To convert cm³ to liters, divide the volume in cm³ by 1000. To convert liters to cm³, multiply the volume in liters by 1000.
9. What are some common mistakes students make when solving cube volume problems?
• Forgetting to cube the side length.
• Incorrectly converting units.
• Confusing volume with surface area.
• Using the wrong formula (e.g., for a cuboid instead of a cube).
10. How is the volume of a cube calculated if only the length of its diagonal is known?
The formula for the volume (V) of a cube given its diagonal (d) is: V = (√3 * d³)/9
11. What are some real-life applications of calculating the volume of a cube?
• Determining the amount of liquid a cubical container can hold.
• Calculating the amount of material needed to construct a cube-shaped object.
• Estimating the space occupied by cubical packages for shipping and storage.
12. Can you give an example of a word problem involving the volume of a cube?
A water tank is cube-shaped with sides of 2 meters. How many liters of water can it hold? (Remember: 1 cubic meter = 1000 liters)
