Learning Objectives
- understand how a frustum is formed;
- calculate frustum volume by subtraction;
- use similarity to find missing dimensions;
- solve cone and pyramid frustum problems;
- combine several mensuration ideas in one problem.
Introduction
This lesson develops Frustums and Advanced Mensuration 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.
What Is a Frustum?
A frustum is formed when the top of a cone or pyramid is cut off by a plane parallel to the base.
Frustum Volume Method
Worked Example
A cone has radius 9 cm and height 15 cm. A smaller similar cone of radius 3 cm and height 5 cm is removed.
Large volume=⅓π(9²)(15)=405π.
Small volume=⅓π(3²)(5)=15π.
Frustum volume=390π cm³.
Using Similarity
If the small and large cones are similar, their corresponding lengths share one scale factor.
Large radius 12 cm, small radius 4 cm, large height 18 cm.
Linear factor small:large = 1:3, so small height = 6 cm.
🧠 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 large cone has volume 500π cm³ and the removed cone has volume 32π cm³. Find frustum volume.
- A pyramid frustum is formed by removing a 40 cm³ pyramid from a 220 cm³ pyramid. Find remaining volume.
- A cone has radius 8 cm, height 12 cm. A similar cone of radius 4 cm is removed. Find the small height and frustum volume.
- A square pyramid has base side 12 cm and height 18 cm. A similar top pyramid with base side 4 cm is removed. Find frustum volume.
- A frustum comes from a cone of radius 10 cm and height 24 cm. The top radius is 6 cm. Find frustum height and volume.
✅ Check Your Work
Show practice answers
- 468π cm³.
- 180 cm³.
- Small height=6 cm. Large volume=256π; small=32π; frustum=224π cm³.
- Scale factor=1/3, small height=6 cm. Large volume=864 cm³; small=32 cm³; frustum=832 cm³.
- Small:large radius=6:10=3:5, so small height=14.4 cm. Frustum height=9.6 cm. Volume=800π−172.8π=627.2π 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
- A frustum is a large solid with a similar smaller top removed.
- Use similarity to find missing lengths.
- Subtract small volume from large volume.