Use a Cubed Radius for Sphere Volume
Sphere volume is (4/3)πr³.
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 sphere has radius 6 cm. Find exact volume. Candidate answers: (A) 144π cm³ (B) 216π cm³ (C) 288π cm³ (D) 864π cm³.
- Independent check inputs: A sphere has radius 9 cm. Find exact volume. Candidate answers: (A) 2916π cm³ (B) 324π cm³ (C) 729π cm³ (D) 972π 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. 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.
Here is the method on the supplied teaching case. Use V=(4/3)πr³: (4/3)π(6)³=288π cm³. The formula cubes radius because this is three-dimensional volume.
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.
- 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 sphere has radius 6 cm. Find exact volume. Candidate answers: (A) 144π cm³ (B) 216π cm³ (C) 288π cm³ (D) 864π cm³. Solution: Use V=(4/3)πr³: (4/3)π(6)³=288π cm³. The formula cubes radius because this is three-dimensional volume. The final answer is 288π cm³. 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.
Check your reasoning
Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.
Helpful tips
- Sphere volume is (4/3)πr³.
- Sphere surface area uses 4πr², a different formula and unit.
See the answer and explanation
972π cm³
Use V=(4/3)πr³: (4/3)π(9)³=972π cm³. The formula cubes radius because this is three-dimensional volume.
Take it further
Changed-input task: A sphere has radius 27 cm. Find exact volume. Candidate answers: (A) 19683π cm³ (B) 26244π cm³ (C) 2916π cm³ (D) 78732π cm³. Show the mathematical reason, not only the selected answer. Expected result: 26244π cm³. Use V=(4/3)πr³: (4/3)π(27)³=26244π cm³. The formula cubes radius because this is three-dimensional volume.
Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.
More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
Your choices are temporary. Feedback appears after you check all the answers.
Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.
Check the subject sources
These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.