g10-10-04 · Syllabus supplement
The same base and vertical height
Which height determines a prism or cone volume?
1 MIN · PREDICT ON PAPER
For base area12 and height5, predict the prism and cone volumes.
Discuss or write first. No typed answer is required to continue.
Use perpendicular height, not a sloping edge.
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
Vprism=Sh andVcone=Sh/3 use the perpendicular height. The one-third relation is a geometric theorem, not a result proved by a fill animation.
Conditions: S>0 and h>0 for a solid; h=0 is degenerate and has zero volume. Use cubic units.
Mathematical justification: For a pyramid, a cross-section at distance z from the apex has area S(z/h)²; summing thin slices gives Sh/3. A prism has constant cross-sectional area S, giving Sh.
1 MIN · EXIT CHECK
Find the cone volume when S=12,h=5.
Check after answering
20 cubic units.
Teacher guidance and extension
Before: Plane area, Pythagoras and units.
Ask: Would leaning the side change the perpendicular height shown?
Watch for: Using a sloping edge as height or reporting square units for volume.
Challenge: Explain why equal heights alone do not fix volumes if base areas differ.
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