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Grade 10 · unit review

Review: 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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A cylinder has radius 7 cm and height 8 cm. What is its exact volume?
Need a hint?
  • A cylinder with a constant circular cross-section has volume equal to base area times perpendicular height.
  • Use radius rather than diameter in πr²h.
2. A cone has radius 7 cm and perpendicular height 24 cm. Find exact volume.
Need a hint?
  • 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.
3. A sphere has radius 21 cm. Find exact volume.
Need a hint?
  • Sphere volume is (4/3)πr³.
  • Sphere surface area uses 4πr², a different formula and unit.
4. A solid has volume 35 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume.
Need a hint?
  • If all corresponding lengths change by a factor k, base areas change by k² and heights by k.
  • The cube rule needs similarity in all dimensions.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. 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. 2. 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. 3. A sphere has radius 27 cm. Find exact volume. Candidate answers: (A) 19683π cm³ (B) 26244π cm³ (C) 2916π cm³ (D) 78732π cm³. 4. A solid has volume 45 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 1215 cm³ (B) 135 cm³ (C) 405 cm³ (D) 48 cm³.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 810π cm³. The circular base has area πr²=81π cm². Multiply by height 10 to get 810π cm³. Radius and diameter are different inputs. 2. 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. 3. 26244π cm³. Use V=(4/3)πr³: (4/3)π(27)³=26244π cm³. The formula cubes radius because this is three-dimensional volume. 4. 1215 cm³. Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 45 by 27 to obtain 1215 cm³. Area would instead scale by 9.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.