g10-10-05 · Syllabus supplement
A frustum is a difference of cones
Can the volume use the average of the two end areas?
1 MIN · PREDICT ON PAPER
Compare the correct volume with hπ(R²+r²)/2 for R=3,r=1,h=4.
Discuss or write first. No typed answer is required to continue.
The missing cross-term matters; averaging end areas overestimates.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Subtract similar cones to obtain V=πh(R²+Rr+r²)/3. Simply averaging end areas omits the way intermediate cross-sections vary.
Conditions: R>0,0≤r≤R,h>0. r=0 gives a cone; r=R gives the cylinder limit.
Mathematical justification: For r<R, similarity gives total cone height H=hR/(R−r). Subtract πR²H/3−πr²(H−h)/3 and factor R³−r³. The result extends continuously to r=R.
1 MIN · EXIT CHECK
Find the volume for R=3,r=1,h=4.
Check after answering
52π/3 cubic units.
Teacher guidance and extension
Before: Plane area, Pythagoras and units.
Ask: Which missing cross-term appears after factoring the difference of cubes?
Watch for: Using the trapezium area rule directly for volume.
Challenge: Show that the average-end-area estimate is too large when R≠r.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore