Solid geometry: sections, limits and design evidence
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 minutes, as an unvalidated planning estimate. This collection contains 8 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Use labeled diagrams or precise coordinate descriptions, and state geometric and modeling assumptions.
Fresh public tasks require written review; saving a response is not verified understanding.
Retain π and radicals for exact values; justify any decimal tolerance or bound.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
In the cube 0≤x,y,z≤3, describe the section x+y+z=2 and prove its shape. Then describe the section in the vertical plane x=y. Explain why a perspective outline alone would not settle either answer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Rotate the filled rectangle 2≤x≤5,0≤z≤3 first about the z-axis and then about the x-axis. Describe sections, dimensions and exact volumes. Does rotating only its boundary fill the same interiors?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A radius-2 circle has an inscribed regular 12-gon of side √6−√2 and apothem (√6+√2)/2. Derive its perimeter and area, derive the corresponding tangent polygon perimeter, and bracket circle area. What further argument gives exact area πr²?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A right prism has a right-triangle base with legs 3 cm and 8 cm and height 7 cm. An oblique prism has rectangular base 3 cm by 4 cm and perpendicular height 7 cm. Compare volumes using complete section rules. Why would agreement at only three heights be insufficient?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A square pyramid has base side 5 cm and height 12 cm. A right cone has radius 5 cm and slant length 13 cm. Find both volumes. For the pyramid use stack gap Bh/n to choose an integer n guaranteeing a gap smaller than 1/2 cm³, and explain the limiting one-third factor.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For a sphere of radius 6 cm, construct the cylinder-minus-double-cone comparison. At central height z=2 cm calculate both section areas, then prove equality at every height and compute sphere volume. Why is one matching height insufficient?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Synthetic measurements give object radii 2 cm at heights 0 and 3 cm, and radius 0 at height 6 cm. Compare a radius-2,height-6 cylinder with a radius-2,height-3 cylinder joined to a radius-2,height-3 cone. Displacement is 50±2 cm³. Use 3.14159<π<3.14160 to assess both models and state needed further evidence.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare closed boxes A (6×10×20 cm) and B (10×10×12 cm), with last dimension designated height. Require capacity ≥ 1200 cm³, height ≤ 20 cm and area ≤ 760 cm². Sheet area density is 0.15 g/cm². Calculate volumes, areas and ideal masses; choose a feasible lighter design, then reassess with height ≤ 15 cm. Could volume density replace area density without more data?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
In the cube 0≤x,y,z≤3, describe the section x+y+z=2 and prove its shape. Then describe the section in the vertical plane x=y. Explain why a perspective outline alone would not settle either answer.
Rotate the filled rectangle 2≤x≤5,0≤z≤3 first about the z-axis and then about the x-axis. Describe sections, dimensions and exact volumes. Does rotating only its boundary fill the same interiors?
A radius-2 circle has an inscribed regular 12-gon of side √6−√2 and apothem (√6+√2)/2. Derive its perimeter and area, derive the corresponding tangent polygon perimeter, and bracket circle area. What further argument gives exact area πr²?
A right prism has a right-triangle base with legs 3 cm and 8 cm and height 7 cm. An oblique prism has rectangular base 3 cm by 4 cm and perpendicular height 7 cm. Compare volumes using complete section rules. Why would agreement at only three heights be insufficient?
A square pyramid has base side 5 cm and height 12 cm. A right cone has radius 5 cm and slant length 13 cm. Find both volumes. For the pyramid use stack gap Bh/n to choose an integer n guaranteeing a gap smaller than 1/2 cm³, and explain the limiting one-third factor.
For a sphere of radius 6 cm, construct the cylinder-minus-double-cone comparison. At central height z=2 cm calculate both section areas, then prove equality at every height and compute sphere volume. Why is one matching height insufficient?
Synthetic measurements give object radii 2 cm at heights 0 and 3 cm, and radius 0 at height 6 cm. Compare a radius-2,height-6 cylinder with a radius-2,height-3 cylinder joined to a radius-2,height-3 cone. Displacement is 50±2 cm³. Use 3.14159<π<3.14160 to assess both models and state needed further evidence.
Compare closed boxes A (6×10×20 cm) and B (10×10×12 cm), with last dimension designated height. Require capacity ≥ 1200 cm³, height ≤ 20 cm and area ≤ 760 cm². Sheet area density is 0.15 g/cm². Calculate volumes, areas and ideal masses; choose a feasible lighter design, then reassess with height ≤ 15 cm. Could volume density replace area density without more data?
