Volume of Composite Solids
Calculate the volume of prisms, cylinders, and complex composite shapes by breaking them into simpler parts.
Volume Formulas for Common Solids
Volume is the amount of 3D space a solid occupies. It is measured in cubic units (cm³, m³, mm³).
Rectangular Prism (Cuboid)
V = l × w × h
length × width × height
Example: l = 5, w = 3, h = 4
V = 5 × 3 × 4 = 60 cm³
Cylinder
V = πr²h
pi × radius squared × height
Example: r = 3, h = 10
V = π × 3² × 10 = 90π ≈ 283 cm³
Triangular Prism
V = ½bhl
half × base × triangle height × length
Example: b = 6, h = 4, l = 10
V = 0.5 × 6 × 4 × 10 = 120 cm³
Any Prism
V = A × h
cross-sectional area × height (or length)
A is the area of the cross-section (base shape).
Composite Solids
A composite solid is made up of two or more simpler 3D shapes joined together. To find the total volume:
Adding Volumes
When parts are joined together, add their volumes.
Vtotal = Vpart 1 + Vpart 2
e.g. A house shape = rectangular prism + triangular prism (roof)
Subtracting Volumes
When a part is removed (like a hole), subtract that volume.
Vtotal = Vwhole − Vremoved
e.g. A hollow cylinder = outer cylinder − inner cylinder
Composite Solid: Prism + Half Cylinder
Rectangular Prism
8 × 4 × 3 = 96 cm³
Half Cylinder
½ × π × 2² × 8 ≈ 50.3 cm³
Total Volume
96 + 50.3 ≈ 146.3 cm³
Strategy for Finding Volume of Composite Solids
- 1 Identify the individual shapes that make up the composite solid.
- 2 Find the measurements needed for each part from the diagram.
- 3 Calculate the volume of each part separately using the appropriate formula.
- 4 Add or subtract the individual volumes to find the total.
Key Vocabulary
Volume
The amount of three-dimensional space occupied by a solid, measured in cubic units (cm³, m³).
Composite Solid
A 3D shape made from two or more simpler shapes joined together or one removed from another.
Cross-Section
The shape formed when a solid is cut through parallel to its base. Area of cross-section × height = volume of prism.
Prism
A 3D shape with two identical parallel bases and rectangular sides. Volume = Area of base × height.
Worked Examples
A swimming pool is a rectangular prism 12 m long, 5 m wide and 1.8 m deep. Find its volume.
V = l × w × h = 12 × 5 × 1.8 = 108 m³
Answer: 108 m³
A cylindrical tin has radius 4 cm and height 10 cm. Find the volume (use π = 3.14).
V = πr²h = 3.14 × 4² × 10 = 3.14 × 16 × 10 = 502.4 cm³
Answer: 502.4 cm³
A composite solid has a rectangular prism (4×3×5 cm) with a cylinder (r=1, h=3 cm) removed. Find the remaining volume.
Vprism = 4 × 3 × 5 = 60 cm³
Vcylinder = π × 1² × 3 = 3π ≈ 9.42 cm³
Vremaining = 60 − 9.42 = 50.58 cm³
Answer: approximately 50.6 cm³
Knowledge Check
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Key Concepts Summary
- ●Rectangular prism: V = l × w × h.
- ●Cylinder: V = πr²h.
- ●Any prism: V = Area of cross-section × height.
- ●Composite solid: split into simpler parts, then add or subtract their volumes.
- ●Volume is always measured in cubic units (cm³, m³, mm³).