Learning Objectives
- calculate volume of prisms;
- calculate volumes of cones and pyramids;
- identify perpendicular height;
- compare solids with the same base and height;
- solve missing-dimension problems.
Introduction
This lesson develops Prisms, Cones and Pyramids 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.
Prisms
A prism has the same cross-section all the way through.
Worked Example: Triangular Prism
Triangle base 6 cm, height 4 cm, prism length 10 cm.
Cross-section area=½×6×4=12 cm².
Volume=12×10=120 cm³.
Cones and Pyramids
Clean Comparison Diagram
Worked Example: Cone
Radius 6 cm, height 9 cm.
V=⅓π(6²)(9)=108π≈339.3 cm³.
Worked Example: Square Pyramid
Square base side 8 m, perpendicular height 12 m.
Base area=64 m², so V=⅓×64×12=256 m³.
🧠 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.
Practice Questions
Foundation- A prism has cross-sectional area 18 cm² and length 14 cm. Find volume.
- Find the volume of a cone with radius 3 cm and height 8 cm.
- Find the volume of a square pyramid with base side 6 m and height 10 m.
- A triangular prism has triangular base 8 cm, triangle height 5 cm and length 12 cm. Find volume.
- A cone has volume 96π cm³ and radius 4 cm. Find height.
- A pyramid has volume 210 cm³ and perpendicular height 10 cm. Find base area.
- A cylinder and cone have the same radius 5 cm and height 12 cm. Find the difference in volume.
✅ Check Your Work
Show practice answers
- 252 cm³.
- 24π≈75.4 cm³.
- ⅓×36×10=120 m³.
- Cross-section=20 cm²; volume=240 cm³.
- 96π=⅓π×16×h → 96=16h/3 → h=18 cm.
- 210=⅓×B×10 → B=63 cm².
- Cylinder=300π; cone=100π; difference=200π≈628.3 cm³.
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
- Prism volume = cross-section area × length.
- Cone and pyramid volumes include a factor of one-third.
- Use perpendicular height.
- Same base and height: cone/pyramid volume is one-third of matching prism.