Stack a Circular Base through a Height
A cylinder with a constant circular cross-section has volume equal to base area times perpendicular height.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A cylinder has radius 2 cm and height 3 cm. What is its exact volume? Candidate answers: (A) 12π cm³ (B) 24π cm³ (C) 4.0π cm³ (D) 48π cm³.
- Independent check inputs: A cylinder has radius 3 cm and height 4 cm. What is its exact volume? Candidate answers: (A) 12.0π cm³ (B) 144π cm³ (C) 36π cm³ (D) 72π cm³.
Learn the idea
Three-dimensional formulas relate solid shapes to base areas, heights and radii. Similar solids change every corresponding length together, making volume respond to the cube of a scale factor rather than its first power. A cylinder with a constant circular cross-section has volume equal to base area times perpendicular height. The squared radius belongs to the base; height supplies the third dimension.
Here is the method on the supplied teaching case. The circular base has area πr²=4π cm². Multiply by height 3 to get 12π cm³. Radius and diameter are different inputs.
Use radius rather than diameter in πr²h. A slanted measurement is not the perpendicular height; exact volume may remain a multiple of π and must use cubic units.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A cylinder has radius 2 cm and height 3 cm. What is its exact volume? Candidate answers: (A) 12π cm³ (B) 24π cm³ (C) 4.0π cm³ (D) 48π cm³. Solution: The circular base has area πr²=4π cm². Multiply by height 3 to get 12π cm³. Radius and diameter are different inputs. The final answer is 12π cm³. Use radius rather than diameter in πr²h. A slanted measurement is not the perpendicular height; exact volume may remain a multiple of π and must use cubic units.
Check your reasoning
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Helpful tips
- A cylinder with a constant circular cross-section has volume equal to base area times perpendicular height.
- Use radius rather than diameter in πr²h.
See the answer and explanation
36π cm³
The circular base has area πr²=9π cm². Multiply by height 4 to get 36π cm³. Radius and diameter are different inputs.
Take it further
Changed-input task: A cylinder has radius 9 cm and height 10 cm. What is its exact volume? Candidate answers: (A) 1620π cm³ (B) 270.0π cm³ (C) 3240π cm³ (D) 810π cm³. Show the mathematical reason, not only the selected answer. Expected result: 810π cm³. The circular base has area πr²=81π cm². Multiply by height 10 to get 810π cm³. Radius and diameter are different inputs.
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