g10-10-03 · Syllabus supplement
A cone height is not its slant height
Which length belongs in the lateral-area formula?
1 MIN · PREDICT ON PAPER
For radius3 and vertical height4, predict the slant length and total surface area of a cone.
Discuss or write first. No typed answer is required to continue.
Use the unwrapped length: slant height for a cone, vertical height for a cylinder.
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
A cylinder unwraps to a rectangle of width2πr and height h. A cone unwraps to a sector with radius ℓ=√(r²+h²) and arc2πr, so its lateral area isπrℓ.
Conditions: r>0,h>0. Total surface includes the base: one for a cone,two for a cylinder. Use square units.
Mathematical justification: Cone-sector area is half its radius times arc length: (1/2)ℓ(2πr)=πrℓ. Add the circular baseπr².
1 MIN · EXIT CHECK
Find the total surface area for a cone with r=3,h=4.
Check after answering
ℓ=5; area=15π+9π=24π.
Teacher guidance and extension
Before: Plane area, Pythagoras and units.
Ask: Where is the vertical height after unwrapping the cone?
Watch for: Using vertical height h inπrh for the cone lateral area.
Challenge: Compare the cone and cylinder total areas with the same radius and vertical height.
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
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