Volume of Cube
A cube is a three-dimensional solid where all six faces are identical squares. It has 6 faces, 12 edges, and 8 vertices, and all edges are of equal length.
The volume of a cube tells us the amount of space enclosed within it. Volume is measured in cubic units such as cm³, m³, or mm³.
Since all edges of a cube are equal, the volume formula is straightforward: it is the cube of the edge length. This is why raising a number to the power of 3 is called "cubing" the number.
Volume of a cube is part of the NCERT Class 8 Mensuration chapter. Understanding this concept is essential for solving problems on capacity, storage, and three-dimensional measurements.
What is Volume of Cube?
Definition: The volume of a cube is the total space occupied by the cube. It is calculated by multiplying the edge length by itself three times.
Key Terms:
- Cube: A 3D solid with 6 identical square faces, 12 equal edges, and 8 vertices.
- Edge (a): The length of one side of the cube. All 12 edges are equal.
- Volume: The amount of space inside the cube, measured in cubic units.
- Face: Each flat surface of the cube. A cube has 6 square faces.
- Cubic Units: Units for volume: cm³, m³, mm³, km³, litres (1 litre = 1000 cm³).
Properties of a cube:
- All faces are congruent squares.
- All edges are equal in length.
- All face angles are 90°.
- A cube is a special case of a cuboid where length = breadth = height.
Volume of Cube Formula
Volume of a Cube:
Volume = a³ = a × a × a
Where:
- a = length of one edge of the cube
Related Formulas:
- Edge from volume: a = ³√V (cube root of volume)
- Total Surface Area of Cube: TSA = 6a²
- Lateral Surface Area: LSA = 4a²
- Diagonal of a cube: d = a√3
Unit Conversions:
- 1 m³ = 1,000,000 cm³ = 10⁶ cm³
- 1 litre = 1000 cm³
- 1 m³ = 1000 litres
- 1 cm³ = 1 mL (millilitre)
Derivation and Proof
Why is the volume of a cube = a³?
Volume measures how many unit cubes fit inside a solid.
- Consider a cube with edge length a units.
- Along one edge, we can place a unit cubes in a row.
- Along the adjacent edge, we can place a rows, giving a² unit cubes on the base (one layer).
- We can stack a such layers, one on top of another.
- Total unit cubes = a × a × a = a³.
Example: A cube with edge 3 cm:
- Bottom layer: 3 × 3 = 9 unit cubes
- 3 layers: 9 × 3 = 27 unit cubes
- Volume = 3³ = 27 cm³
Cube as a special cuboid:
A cuboid has volume = length × breadth × height. A cube is a cuboid where l = b = h = a. So volume = a × a × a = a³.
Types and Properties
Problems on volume of cube can be classified as:
1. Finding volume given edge length:
- Directly compute V = a³.
2. Finding edge length given volume:
- Find cube root: a = ³√V.
3. Comparing volumes of two cubes:
- If edge is doubled, volume becomes 8 times (2³ = 8).
4. Unit conversion problems:
- Convert between cm³, m³, litres.
5. Real-world applications:
- Finding capacity of a cubic container.
- Calculating amount of material to fill a cube.
6. Combined with surface area:
- Given surface area, find volume (or vice versa).
Solved Examples
Example 1: Example 1: Basic volume calculation
Problem: Find the volume of a cube with edge 5 cm.
Solution:
Given:
- Edge (a) = 5 cm
Using the formula:
- Volume = a³ = 5³
- = 5 × 5 × 5
- = 125 cm³
Answer: The volume of the cube is 125 cm³.
Example 2: Example 2: Finding edge from volume
Problem: The volume of a cube is 512 cm³. Find the edge length.
Solution:
Given:
- Volume = 512 cm³
Finding the edge:
- a = ³√512
- 512 = 8 × 64 = 8 × 8 × 8 = 8³
- a = 8 cm
Answer: The edge length is 8 cm.
Example 3: Example 3: Volume in litres
Problem: A cubic water tank has an edge of 50 cm. Find its capacity in litres.
Solution:
Step 1: Find volume in cm³
- Volume = 50³ = 50 × 50 × 50 = 1,25,000 cm³
Step 2: Convert to litres
- 1 litre = 1000 cm³
- Capacity = 1,25,000 / 1000 = 125 litres
Answer: The tank can hold 125 litres of water.
Example 4: Example 4: Effect of doubling the edge
Problem: A cube has edge 4 cm. If the edge is doubled, how many times does the volume increase?
Solution:
Original volume:
- V₁ = 4³ = 64 cm³
New edge = 2 × 4 = 8 cm:
- V₂ = 8³ = 512 cm³
Ratio:
- V₂ / V₁ = 512 / 64 = 8 times
General rule: If the edge is multiplied by k, the volume becomes k³ times. For k = 2: 2³ = 8 times.
Answer: The volume increases 8 times.
Example 5: Example 5: Volume from total surface area
Problem: The total surface area of a cube is 294 cm². Find its volume.
Solution:
Step 1: Find edge from surface area
- TSA = 6a²
- 294 = 6a²
- a² = 294 / 6 = 49
- a = √49 = 7 cm
Step 2: Find volume
- V = a³ = 7³ = 7 × 7 × 7 = 343 cm³
Answer: The volume is 343 cm³.
Example 6: Example 6: Filling a cube with smaller cubes
Problem: How many cubes of edge 2 cm can fit inside a cube of edge 10 cm?
Solution:
Volume of large cube:
- V = 10³ = 1000 cm³
Volume of small cube:
- v = 2³ = 8 cm³
Number of small cubes:
- = 1000 / 8 = 125
Alternative method: Along each edge: 10/2 = 5 cubes. Total = 5 × 5 × 5 = 125.
Answer: 125 small cubes can fit inside.
Example 7: Example 7: Volume in m³
Problem: Find the volume of a cube with edge 1.5 m. Express the answer in m³ and cm³.
Solution:
Volume in m³:
- V = (1.5)³ = 1.5 × 1.5 × 1.5
- = 2.25 × 1.5
- = 3.375 m³
Volume in cm³:
- 1 m³ = 10⁶ cm³
- 3.375 × 10⁶ = 33,75,000 cm³
Answer: Volume = 3.375 m³ = 33,75,000 cm³.
Example 8: Example 8: Cost of painting a cube
Problem: A cube has volume 729 cm³. Find the cost of painting its outer surface at Rs 2 per cm².
Solution:
Step 1: Find edge
- a = ³√729 = 9 cm (since 9³ = 729)
Step 2: Find total surface area
- TSA = 6a² = 6 × 9² = 6 × 81 = 486 cm²
Step 3: Find cost
- Cost = 486 × 2 = Rs 972
Answer: The cost of painting is Rs 972.
Example 9: Example 9: Comparing volumes of two cubes
Problem: The edges of two cubes are 6 cm and 10 cm. Find the ratio of their volumes.
Solution:
- Volume of cube 1 = 6³ = 216 cm³
- Volume of cube 2 = 10³ = 1000 cm³
- Ratio = 216 : 1000 = 27 : 125
Shortcut: Ratio of volumes = (ratio of edges)³ = (6/10)³ = (3/5)³ = 27/125.
Answer: The ratio of volumes is 27 : 125.
Example 10: Example 10: Ice cube melting problem
Problem: 64 ice cubes, each with edge 3 cm, are melted and poured into a larger cubic mould. What is the edge of the larger cube?
Solution:
Volume of each small ice cube:
- = 3³ = 27 cm³
Total volume of 64 cubes:
- = 64 × 27 = 1728 cm³
Edge of larger cube:
- a = ³√1728 = 12 cm (since 12³ = 1728)
Answer: The edge of the larger cube is 12 cm.
Real-World Applications
Packaging and Storage: Boxes, containers, and storage units are often cubic. Calculating volume helps determine how much they can hold.
Construction: Builders calculate volumes of cubic structures, concrete blocks, and foundations. A standard concrete block is measured in cubic units.
Science and Chemistry: Scientists measure volumes of cubic containers in laboratories. One cubic centimetre (cm³) equals one millilitre (mL).
Ice and Cooling: Ice cube trays produce cubes of specific sizes. Knowing the volume helps calculate the weight of ice (since density of ice ≈ 0.92 g/cm³).
Shipping: Cargo containers and packages are measured in cubic metres or cubic feet. Shipping costs often depend on the volume of goods.
Mathematics: The concept of "cubing" a number (raising to the power 3) comes directly from the volume of a cube with that edge length.
Key Points to Remember
- A cube has 6 equal square faces, 12 equal edges, and 8 vertices.
- Volume of a cube = a³, where a is the edge length.
- To find edge from volume: a = ³√V (cube root).
- Volume is measured in cubic units: cm³, m³, mm³.
- 1 litre = 1000 cm³ and 1 m³ = 1000 litres.
- If the edge is doubled, volume becomes 8 times. If tripled, 27 times. In general, k times edge gives k³ times volume.
- A cube is a special cuboid where all dimensions are equal.
- The ratio of volumes of two cubes = (ratio of their edges)³.
- Total surface area of a cube = 6a². This can be used to find edge, and then volume.
- Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 are volumes of cubes with edges 1 through 10.
Practice Problems
- Find the volume of a cube with edge 12 cm.
- The volume of a cube is 1331 cm³. Find the edge length.
- A cubic tank has an edge of 80 cm. How many litres of water can it hold?
- If the edge of a cube is tripled, how many times does the volume increase?
- The total surface area of a cube is 600 cm². Find its volume.
- How many cubes of edge 5 cm can be cut from a cube of edge 30 cm?
- A cube has volume 2744 cm³. Find its total surface area.
- Two cubes have edges 3 cm and 4 cm. If they are melted and recast into a single cube, find the edge of the new cube.
Frequently Asked Questions
Q1. What is the formula for volume of a cube?
Volume of a cube = a³ = a × a × a, where a is the length of one edge. All edges of a cube are equal.
Q2. What are the units of volume?
Volume is measured in cubic units: cm³ (cubic centimetres), m³ (cubic metres), mm³, etc. 1 cm³ = 1 mL, and 1000 cm³ = 1 litre.
Q3. How do you find the edge of a cube from its volume?
Take the cube root: edge = ³√V. For example, if V = 343 cm³, then edge = ³√343 = 7 cm.
Q4. What is the difference between a cube and a cuboid?
A cube has all edges equal (l = b = h = a), so Volume = a³. A cuboid has different length, breadth, and height, so Volume = l × b × h. A cube is a special case of a cuboid.
Q5. How does volume change when the edge doubles?
When edge doubles from a to 2a, volume changes from a³ to (2a)³ = 8a³. Volume becomes 8 times the original.
Q6. What is 1 cubic metre in litres?
1 m³ = 1000 litres. This is because 1 m = 100 cm, so 1 m³ = 100³ cm³ = 10,00,000 cm³, and 10,00,000 / 1000 = 1000 litres.
Q7. How do you find volume if surface area is given?
Total surface area = 6a². Find a² = TSA/6, then a = √(TSA/6). Then Volume = a³.
Q8. How many small cubes fit in a large cube?
Divide the volume of the large cube by the volume of the small cube. Or: along each edge, (large edge / small edge) cubes fit. Total = (large edge / small edge)³.
Q9. What is the diagonal of a cube?
The space diagonal of a cube with edge a is d = a√3. The face diagonal is a√2.
Q10. Why is raising a number to the power 3 called cubing?
Because a³ gives the volume of a cube with edge a. Just as a² is called 'squaring' (area of a square with side a), a³ is called 'cubing'.
