SME
Grade 10 Math · Geometry
Surface area and volume of solids, cross sections and Cavalieri's principle
Use this for “Find the volume of a cylinder with radius 3 cm and height 8 cm,” and why stacked slices give it.
Diagram
Both stacks are 12 coins of radius 3 cm, 8 cm tall; one leans.
At every height the two stacks cut a coin of the same area, π × r².
So they hold the same volume (Cavalieri's principle): V = π × 3² × 8 = 72π ≈ 226.19 cm³
This is an example. Type your own values to start a new problem.
V (cm³)
=
π
×
r
2
×
h
Formulas
V
=
π
×
r
2
×
h
72π
=
π
×
3
2
×
8
Assumptions
•
A cylinder is a stack of circles, each with area πr², so V = πr²h.
•
Cavalieri: solids with equal cross-sections at every height have equal volumes.
•
So a leaning stack keeps the same volume as a straight one.
Step-by-step
WE KNOW
r = 3 cm, h = 8 cm
FIND
volume (V)
Step 1 · Find volume (V)
Use V = π × r² × h
The base’s area πr² times the height.
V
=
π
×
3
2
×
8
V
=
9
×
8
×
π
V
=
72
×
π
V
=
72π
cm³
(≈
226.19
cm³)
Check
Put the numbers back in. Both sides should match.
72π
=
π
×
3
2
×
8
✓
Topic check
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