Home›Calculus 2›The theorem of Pappus›Problem 5.645

The theorem of Pappus

Problem 5.645 · medium

The triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (3, 0) \) and \( \displaystyle (0, 5) \) is rotated about the \( \displaystyle x \)-axis. Use the theorem of Pappus to find the volume.

Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ 1 \cdot \frac{1}{2} \cdot 3 \cdot 5 = \frac{15}{2} \]
Area ½·base·height.✓ Proved
A triangle's centroid is the average of its vertices: ȳ = (0 + 0 + 5)/3 = 5/3.
Reviewed
\[ 2 \pi 5 \cdot \frac{1}{3} \cdot 15 \cdot \frac{1}{2} = 25 \pi \]
V = 2π·d·A.✓ Proved

Answer \( 25 \pi \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the volume by disks or shells, numerically

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid's y-coordinate (distance to the x-axis) is correctly identified as 5/3. The final volume calculation is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid's y-coordinate (distance to the x-axis) is correctly identified as 5/3. The final volume calculation is accurate.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the centroid's y-coordinate (distance to the x-axis) is correctly identified as 5/3, and the final volume calculation is accurate.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the y-coordinate of the centroid is correctly identified as the distance to the x-axis, and the final volume calculation is accurate.
gpt-oss:20b: pass 2026-10-15

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/pappus, checked 2026-10-18 with SymPy 1.14.0.