The theorem of Pappus
Problem 5.620 · medium
The triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (5, 0) \) and \( \displaystyle (0, 1) \) 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
- \[ 1 \cdot \frac{1}{2} \cdot 5 \cdot 1 = \frac{5}{2} \]Area ½·base·height.✓ Proved
- A triangle's centroid is the average of its vertices: ȳ = (0 + 0 + 1)/3 = 1/3.Reviewed
- \[ 2 \pi 1 \cdot \frac{1}{3} \cdot 5 \cdot \frac{1}{2} = \frac{5 \pi}{3} \]V = 2π·d·A.✓ Proved
Answer \( \frac{5 \pi}{3} \)
✓ 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: passqwen3.6:27b-mlx: pass — 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 1/3, and 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, the centroid's y-coordinate (distance to the x-axis) is correctly identified as 1/3, and the final volume calculation is accurate.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the centroid y-coordinate is correctly identified as the average of the y-coordinates of the vertices, and the final volume calculation is accurate.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the area and the y-coordinate of the centroid. The application of Pappus's Second Theorem is accurate, and the final calculation is correct.gpt-oss:20b: pass 2026-10-14
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.