Compare a Cone with Its Matching Cylinder
A cone’s volume is one third of a cylinder with the same base area and 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 cone has radius 2 cm and perpendicular height 9 cm. Find exact volume. Candidate answers: (A) 12π cm³ (B) 24π cm³ (C) 36π cm³ (D) 6.0π cm³.
- Independent check inputs: A cone has radius 3 cm and perpendicular height 12 cm. Find exact volume. Candidate answers: (A) 108π cm³ (B) 18.0π 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 cone’s volume is one third of a cylinder with the same base area and perpendicular height. That relation makes the factor one third part of the whole base-times-height product.
Here is the method on the supplied teaching case. Cone volume is one third of the matching cylinder: (1/3)π(2)²(9)=12π cm³. Use perpendicular height, not slant height.
Keep the perpendicular height separate from the cone’s slant height. Multiplying by the same base and height without dividing by three would compute the matching cylinder, not the cone.
- 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 cone has radius 2 cm and perpendicular height 9 cm. Find exact volume. Candidate answers: (A) 12π cm³ (B) 24π cm³ (C) 36π cm³ (D) 6.0π cm³. Solution: Cone volume is one third of the matching cylinder: (1/3)π(2)²(9)=12π cm³. Use perpendicular height, not slant height. The final answer is 12π cm³. Keep the perpendicular height separate from the cone’s slant height. Multiplying by the same base and height without dividing by three would compute the matching cylinder, not the cone.
Check your reasoning
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Helpful tips
- A cone’s volume is one third of a cylinder with the same base area and perpendicular height.
- Keep the perpendicular height separate from the cone’s slant height.
See the answer and explanation
36π cm³
Cone volume is one third of the matching cylinder: (1/3)π(3)²(12)=36π cm³. Use perpendicular height, not slant height.
Take it further
Changed-input task: A cone has radius 9 cm and perpendicular height 30 cm. Find exact volume. Candidate answers: (A) 1620π cm³ (B) 2430π cm³ (C) 405.0π cm³ (D) 810π cm³. Show the mathematical reason, not only the selected answer. Expected result: 810π cm³. Cone volume is one third of the matching cylinder: (1/3)π(9)²(30)=810π cm³. Use perpendicular height, not slant height.
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