Find the volume of a rectangular prism 7×8×12.
Answer: 672
Reasoning: V=lwh=672.
Reasoning: V=lwh=672.
Unit 11 · Lesson 03 · Solids, Surface Area & Volume
Spheres: Surface Area and Volume · Teacher Key
Use nets, cross-sections, Cavalieri’s principle, and dimensional reasoning to calculate and compare surface area and volume.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to use nets, cross-sections, Cavalieri’s principle, and dimensional reasoning to calculate and compare surface area and volume.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Use nets, cross-sections, Cavalieri’s principle, and dimensional reasoning to calculate and compare surface area and volume.
V=BhV=\frac13BhSA_{cyl}=2\pi r^2+2\pi rhV_{sphere}=\frac43\pi r^3
Worked reasoning
Process: Sketch or inspect the net, identify every exposed surface, calculate each contribution, then compare with a volume calculation using cubic units.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: Surface area counts exposed two-dimensional faces, while volume measures three-dimensional capacity; their units and scale behavior differ.
Khaemenes Academy · Open-Age Geometry
Spheres: Surface Area and Volume
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Find the volume of a cylinder with radius 6 and height 7 in exact form.
Answer: 252π
Reasoning: V=πr²h=252π.
Reasoning: V=πr²h=252π.
Find the volume of a cone with radius 6 and height 7 in exact form.
Answer: 84π
Reasoning: V=1/3πr²h.
Reasoning: V=1/3πr²h.
Find the surface area of a cube with side 7.
Answer: 294
Reasoning: Six congruent square faces: 6·7².
Reasoning: Six congruent square faces: 6·7².
A solid is enlarged by scale factor 2. By what factor does volume change?
Answer: 8
Reasoning: Volume scales by k³.
Reasoning: Volume scales by k³.
What cross-section results from slicing a cylinder parallel to its circular bases?
Answer: A circle
Reasoning: A parallel slice has the same shape as the base.
Reasoning: A parallel slice has the same shape as the base.