SME
Statics · Problem type
Volume by Pappus’s theorem
Use this for “A 2 × 4 cm rectangle turns about an axis 5 cm from its centroid. Find the volume of the ring.”
Diagram
The region under y = 4 on [4, 6], turned about the y-axis.
Its area is A = 8 cm², with its centroid r̄ = 5 cm from the axis.
The centroid travels a circle 2πr̄ = 2π × 5 = 31.42 cm.
Pappus: V = 2πr̄A = 2π × 5 × 8 = 251.3 cm³.
Rectangle width b = 2 cm
This is an example. Type your own values to start a new problem.
A (cm²)
=
b (cm)
×
h (cm)
V (cm³)
=
2
×
π
×
r̄ (cm)
×
A (cm²)
Formulas
A
=
b
×
h
8 = 2 × 4
V
=
2
×
π
×
r̄
×
A
251.3274 = 2 × π × 5 × 8
Assumptions
•
Pappus: a flat area turned a full circle about an axis in its plane sweeps V = 2πr̄A.
•
r̄ is the distance from the axis to the area’s centroid; the area must not cross the axis.
•
Here the area is a b × h rectangle with its side h parallel to the axis.
Step-by-step
WE KNOW
b = 2 cm, h = 4 cm, r̄ = 5 cm
FIND
area (A), volume (V)
Step 1 · Find area (A)
Use A = b × h
The area of the rectangle is its width times its height.
A
=
2
×
4
A
=
8
cm²
Step 2 · Find volume (V)
Use V = 2 × π × r̄ × A
The centroid travels a circle 2πr̄ long, and the area sweeps that far.
V
=
2
×
π
×
5
×
8
V
=
10
×
8
×
π
V
=
80
×
π
V
=
251.3274
cm³
Check
Put the numbers back in. Both sides should match.
8 = 2 × 4 ✓
251.3274 = 2 × π × 5 × 8 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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