Goal and prerequisites
Use cone, sphere and composite-solid formulas while matching length, area and volume scale factors. Review mensuration before starting.
Method and its conditions
A cone has volumeπr²h/3, a sphere has surface area 4πr², and a hemisphere has half a sphere’s volume. For similar solids with length factor k, area factor is k² and volume factor is k³. Identify the dimension before applying a scale factor.
Worked example: show the decisions
Similar solids with volume ratio 27: 1 have length ratio 3: 1 and area ratio 9: 1. If the smaller area is 8, the larger is 72. A cylinder of radius 3 and height 6 topped by a hemisphere of radius 3 has volume 54π+18π=72π.
Contrast: diagnose an incorrect approach
A joined internal circle is not part of the exposed surface, although its radius still determines both volumes. For volume, add non-overlapping component volumes; for surface area, remove hidden contact surfaces. A volume factor does not multiply a corresponding length directly.
Transfer: try a changed situation
A cone has radius 4 and perpendicular height 9. Give its volume exactly, and find the surface-area factor for a similar cone with twice every length.
Transfer solution and interpretation
Volume isπ×16×9/3=48π cubic units. Doubling lengths multiplies surface area by 4 and volume by 8. The perpendicular height belongs in the volume formula; slant height is a different measurement.
A cone has radius 3 cm and perpendicular height 5 cm. Its volume is cπ cm³. Find c.
- Use cone volumeπr²h/3 and report only the coefficient ofπ.
- Using the stated quantities: 3^2×5/3 = 15.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027