Learning Objectives
- split composite shapes and solids into familiar parts;
- add and subtract areas or volumes;
- identify hidden and exposed surfaces;
- solve hollow-solid problems;
- check whether the final result is sensible.
Introduction
This lesson develops Composite Shapes and Solids from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
Composite Shapes
A composite object is made from two or more familiar shapes or solids.
Composite Area
Composite Volume
A solid consists of a cylinder of radius 4 cm and height 10 cm topped by a cone of the same radius and height 6 cm.
Cylinder volume=π(4²)(10)=160π.
Cone volume=⅓π(4²)(6)=32π.
Total volume=192π cm³.
Hollow Solids
Subtract the inner volume from the outer volume.
A hollow cylinder has outer radius 6 cm, inner radius 4 cm and height 10 cm.
Volume of material=π(6²−4²)(10)=200π cm³.
Surface Area of Joined Solids
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Worked Examples
Reason through the method
- Identify the known information and what must be found.
- Choose the definition, property or formula that connects them.
- Substitute carefully and show each step.
- Check the notation, units and whether the result is sensible.
Practice Questions
Foundation- Find the area of a 12 cm by 8 cm rectangle with a semicircle of diameter 8 cm attached.
- A solid combines a cuboid 10×6×4 cm and a cube of side 3 cm. Find total volume.
- A cylinder of radius 5 cm and height 12 cm has a hemisphere of radius 5 cm attached. Find total volume.
- A hollow cuboid has outer dimensions 12×10×8 cm and inner cavity 10×8×6 cm. Find volume of material.
- Two cubes of side 5 cm are joined face-to-face. Find total volume and surface area.
✅ Check Your Work
Show practice answers
- Rectangle=96; semicircle=½π(4²)=8π; total=96+8π≈121.1 cm².
- 240+27=267 cm³.
- Cylinder=300π; hemisphere=⅔π(5³)=250π/3; total=1150π/3≈1204.3 cm³.
- Outer=960; inner=480; material=480 cm³.
- Volume=250 cm³. Surface area=2×150−2×25=250 cm².
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Split composite objects into familiar parts.
- Add attached parts and subtract removed parts.
- Joined faces are not exposed.
- Hollow volume = outer − inner.