The decision and its conditions
Surface area counts exposed faces; volume counts filled space. A closed rectangular prism has three pairs of matching rectangular faces. For inverse volume questions, isolate the missing dimension before calculating. A cone occupies one third of a cylinder with the same base and height; the formula must include that factor. When two nonoverlapping solids are joined, their volumes add, but their total exposed surface area does not simply add if faces touch. Keep square and cubic units visible and identify whether a container has a lid.
Worked example
A closed rectangular box measures 4 by 2 by 3 cm. What is its total surface area in cm²? A closed box has three pairs of matching faces. One face of each pair has areas 8, 12, and 6 cm². Double their sum: 2×(8+12+6)=52 cm².
A contrasting decision
A cylinder has radius 3 cm and volume 54π cm³. Using V=πr²h, find its height. Cancel π to get 54=3²h. The base-area coefficient is 9. Height is 54/9=6 cm.
Diagnose a competing method
The first example’s tempting answer 26 follows this mistake: Count each face size once rather than its matching pair. Count faces rather than filled cubic space. There are two of each face size.
Transfer before revealing
A cone has radius 4 cm and height 9 cm. Using V=πr²h/3, express its volume as kπ cm³ and find k. Pause and identify the requested quantity. Solution: Square the radius to get 16. Multiply by height 9 and divide by three. k=16×9/3=48.
Check the boundary of the method
When two nonoverlapping solids are joined, their volumes add, but their total exposed surface area does not simply add if faces touch. Divide volume by base area, keeping the radius squared. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
A closed rectangular box measures 4 by 2 by 3 cm. What is its total surface area in cm²?
- A closed box has three pairs of matching faces.
- One face of each pair has areas 8, 12, and 6 cm².
- Double their sum: 2×(8+12+6)=52 cm².
Compare the other answer choices
Add edge lengths instead of face areas.
Count each face size once rather than its matching pair.
Omit the pair of width-by-height faces.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning Assessment Guide for Educators