What is Volume?
Volume is the amount of three-dimensional (3D) space an object occupies or a container can hold. Think of how much water fills a swimming pool, or how much air is inside a box.
Volume is a three-dimensional measurement. Common units for volume include:
- Cubic millimetres (mm³)
- Cubic centimetres (cm³)
- Cubic metres (m³)
- Millilitres (ml) - often for liquids
- Litres (L) - often for liquids
It's important to note that 1 cm³ = 1 ml.
Remember the difference:
- Area measures a 2D surface (like the amount of paint for a flat wall) and is in square units (cm², m²).
- Volume measures a 3D space (like the amount of water in a bottle) and is in cubic units (cm³, m³).
Quick Recap: Understanding Pi (π) and Circles
When dealing with circles and shapes that include circles (like cylinders), you'll encounter a special number called Pi (π).
Pi (represented by the Greek letter π) is a mathematical constant. It represents the ratio of a circle's circumference to its diameter. This ratio is the same for all circles, no matter their size.
The value of π is approximately:
3.14(to 2 decimal places) - This is commonly used in Functional Skills.3.14159(to 5 decimal places)- As a fraction, it's often approximated as
22/7.
Pi is an irrational number, which means its decimal representation never ends and never enters a permanently repeating pattern.
Key parts of a circle:
- Radius (r): The distance from the center of the circle to any point on its edge.
- Diameter (d): The distance across the circle, passing through its center. The diameter is twice the radius (
d = 2r). - Circumference (C): The distance around the outside of the circle (its perimeter).
Key measurements and formulas involving π for circles:
- Circumference (C) (distance around):
- Formula 1:
C = π × diameter (d)Example: If diameter (d) = 10cm, C ≈
3.14 × 10cm = 31.4cm. - Formula 2:
C = 2 × π × radius (r)Example: If radius (r) = 5cm, C ≈
2 × 3.14 × 5cm = 31.4cm.
- Formula 1:
- Area (A) (surface covered):
- Formula:
A = π × radius² (πr²)Example: If radius (r) = 5cm, A ≈
3.14 × (5cm)² = 3.14 × 25cm² = 78.5cm².
- Formula:
Understanding these formulas is crucial for calculating the volume of cylindrical shapes, as the base of a cylinder is a circle.
Calculating Volume of Basic Shapes
Cube
All sides are equal (length 's').
Volume = side × side × side = s³
Example: If a cube has a side of 3cm, Volume = 3cm × 3cm × 3cm = 27cm³.
Cuboid (Rectangular Prism)
Has length 'l', width 'w', and height 'h'.
Volume = length × width × height
Example: If length = 6m, width = 2m, height = 3m, Volume = 6m × 2m × 3m = 36m³.
Cylinder
'r' is radius of the circular base, 'h' is height, π (pi) ≈ 3.14.
Volume = π × radius² × height (Area of base × height)
Example: If radius = 3cm, height = 10cm, Volume ≈ 3.14 × (3cm)² × 10cm = 3.14 × 9cm² × 10cm = 282.6cm³.
Triangular Prism
Volume = Area of triangular cross-section × length of prism.
Volume = (½ × base of triangle × height of triangle) × length
Example: If triangular face has base 4m, height 3m, and prism length is 10m:
Area of triangle = ½ × 4m × 3m = 6m².
Volume = 6m² × 10m = 60m³.
Common Volume & Capacity Conversions
| From | To | Conversion Factor |
|---|---|---|
| Cubic Centimetres (cm³) | Millilitres (ml) | 1 cm³ = 1 ml |
| Millilitres (ml) | Litres (L) | Divide by 1000 (1000 ml = 1 L) |
| Litres (L) | Millilitres (ml) | Multiply by 1000 |
| Litres (L) | Cubic Centimetres (cm³) | Multiply by 1000 (1 L = 1000 cm³) |
| Cubic Metres (m³) | Litres (L) | Multiply by 1000 (1 m³ = 1000 L) |
| Cubic Metres (m³) | Cubic Centimetres (cm³) | Multiply by 1,000,000 (1 m³ = 100cm × 100cm × 100cm) |
Exam-Style Volume Problems
1. Fish Tank Capacity
A rectangular fish tank is 60cm long, 30cm wide, and 40cm high. How many litres of water can it hold when full? (1 litre = 1000 cm³)
Answer: 72 litres
Explanation:
- Calculate the volume of the cuboid (fish tank):
Volume =length × width × height
Volume =60cm × 30cm × 40cm = 72000 cm³. - Convert cm³ to litres:
72000 cm³ ÷ 1000 cm³/litre = 72 litres.
2. Cylindrical Water Butt
A cylindrical water butt has a radius of 0.5 metres and a height of 1.2 metres. Calculate its volume in cubic metres. (Use π ≈ 3.14)
Answer: Approximately 0.942 m³
Explanation:
- Formula for volume of a cylinder: Volume =
π × radius² × height. - Radius (r) = 0.5m, Height (h) = 1.2m, π ≈ 3.14.
- Calculation: Volume ≈
3.14 × (0.5m)² × 1.2m
Volume ≈3.14 × 0.25m² × 1.2m
Volume ≈0.785m² × 1.2m
Volume ≈0.942 m³.
3. Concrete for a Triangular Prism Ramp
A ramp is in the shape of a triangular prism. The triangular face has a base of 2m and a perpendicular height of 0.5m. The length of the ramp (prism) is 3m. How much concrete, in m³, is needed to make the ramp?
Answer: 1.5 m³
Explanation:
- Calculate the area of the triangular cross-section:
Area =(½ × base × height) = ½ × 2m × 0.5m = 0.5m². - Calculate the volume of the prism:
Volume =Area of cross-section × length
Volume =0.5m² × 3m = 1.5m³.
4. Comparing Cube Volumes
Cube A has a side length of 4cm. Cube B has a side length that is double that of Cube A. How many times larger is the volume of Cube B compared to Cube A?
Answer: Cube B's volume is 8 times larger.
Explanation:
- Volume of Cube A =
side³ = 4cm × 4cm × 4cm = 64 cm³. - Side length of Cube B =
2 × 4cm = 8cm. - Volume of Cube B =
side³ = 8cm × 8cm × 8cm = 512 cm³. - Comparison:
Volume B / Volume A = 512 cm³ / 64 cm³ = 8.
5. Water in a Cylindrical Tank
A cylindrical tank has a diameter of 2 metres and is filled with water to a height of 1.5 metres. What is the volume of water in the tank in litres? (Use π ≈ 3.14 and 1 m³ = 1000 litres).
Answer: Approximately 4710 litres
Explanation:
- Radius (r) = Diameter / 2 =
2m / 2 = 1m. - Height of water (h) = 1.5m.
- Volume of water (cylinder) =
π × r² × h. - Calculation: Volume ≈
3.14 × (1m)² × 1.5m = 3.14 × 1m² × 1.5m = 4.71 m³. - Convert m³ to litres:
4.71 m³ × 1000 L/m³ = 4710 L.
Interactive Volume Calculator
Volume: ? (cubic units)
Key Points for Volume
- Volume measures the 3D space an object occupies.
- Units for volume are cubic units (e.g., cm³, m³). For liquid capacity, ml and L are common (1ml = 1cm³).
- Know the formulas for common 3D shapes. For cylinders, use π ≈ 3.14.
- For prisms, the general formula is: Volume = Area of cross-section × length (or height).
- Always ensure units are consistent before calculation.