Study guide: Geometry: Curved Solids and Volume Scaling
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.
Ideas to explain in your own words
- 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. Use radius rather than diameter in πr²h.
- 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. Keep the perpendicular height separate from the cone’s slant height.
- Sphere volume is (4/3)πr³. The cubic dependence matches three-dimensional scaling: doubling the radius multiplies volume by eight when the sphere stays geometrically similar. Sphere surface area uses 4πr², a different formula and unit.
- If all corresponding lengths change by a factor k, base areas change by k² and heights by k. Their product makes the similar solid’s volume change by k³. The cube rule needs similarity in all dimensions.
Useful vocabulary
- perpendicular height
- The perpendicular distance from a base plane to the opposite base or apex.
- similar solids
- Solids with matching shapes and one common length scale factor.
- cubic scale factor
- The third power of a common length scale.
Watch your reasoning
- 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.
- 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.
- Sphere surface area uses 4πr², a different formula and unit. Cube the radius and include the factor four thirds; an exact multiple of π should not be rounded unless requested.
A worked review example
Complete supplied guide examples: Stack a Circular Base through a Height — 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. The circular base has area πr²=4π cm². Multiply by height 3 to get 12π cm³. Radius and diameter are different inputs. Compare a Cone with Its Matching Cylinder — 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. Cone volume is one third of the matching cylinder: (1/3)π(2)²(9)=12π cm³. Use perpendicular height, not slant height. Use a Cubed Radius for Sphere Volume — A sphere has radius 6 cm. Find exact volume. Candidate answers: (A) 144π cm³ (B) 216π cm³ (C) 288π cm³ (D) 864π cm³. Use V=(4/3)πr³: (4/3)π(6)³=288π cm³. The formula cubes radius because this is three-dimensional volume. Cube a Common Length Scale Factor — A solid has volume 10 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 13 cm³ (B) 270 cm³ (C) 30 cm³ (D) 90 cm³. Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 10 by 27 to obtain 270 cm³. Area would instead scale by 9.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.