Cylinder from diameter
A cylinder with diameter six centimeters and height five has radius three. Its base area is 9π square centimeters, so its volume is 45π cubic centimeters. Squaring the diameter would make the base area four times too large. Omitting the height would return only the area, even if the numerical answer looked plausible.
Similarity and a sphere
A solid with volume ten enlarged by length factor three has volume 270, because the volume factor is 27. The area factor nine answers a different question. For a sphere with radius three, V=(4/3)πr³ gives 36π. The coefficient of π is 36, while the full physical volume still includes π and cubic units.
Subtract regions, not dimensions
A 10-by-8 rectangular floor with a 3-by-2 rectangular opening has remaining area 80−6=74. Using (10−3)(8−2)=42 would remove full strips along the two dimensions, describing a different shape. Establish that the opening lies inside the floor and that the dimensions refer to the removed region.
Transfer and check
A sphere radius doubles. Explain why its volume becomes eight times as large although its surface area becomes four times as large. For a cylinder of diameter eight and height three, find the volume coefficient 48. State each intermediate unit so a mistaken use of circumference becomes visible.
A cylinder has diameter 6 cm and height 5 cm. Its volume is cπ cm³. What is c?
- The radius is half the diameter, 3 cm.
- Volume=πr²h=π(3)²(5).
- The coefficient is 45; volume has cubic units.
Compare the other answer choices
Omit the height.
Use circumference times height rather than circular-base area times height.
Use diameter as radius in the squared-radius formula.
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.
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