Count Both Faces in Each Opposite Pair
A closed rectangular box has three pairs of congruent faces.
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 closed rectangular box is 2 cm by 3 cm by 4 cm. What is its total surface area? Candidate answers: (A) 18 cm² (B) 24 cm² (C) 26 cm² (D) 52 cm².
- Independent check inputs: A closed rectangular box is 3 cm by 4 cm by 5 cm. What is its total surface area? Candidate answers: (A) 24 cm² (B) 47 cm² (C) 60 cm² (D) 94 cm².
Learn the idea
Geometric measures answer different questions: circumference is boundary length, area covers a region, and volume counts space. A formula works only when its input dimensions and output units match the question being asked. A closed rectangular box has three pairs of congruent faces. Compute each different rectangle’s area, add those three areas, and double to account for its opposite face.
Here is the method on the supplied teaching case. Opposite faces occur in pairs. Add 2(2×3+2×4+3×4)=52 cm². All six faces count because the box is closed.
The word closed matters: an open-top box would omit one face. Do not confuse lwh with the sum of face areas; surface material is measured in square units, not cubic units.
- 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 closed rectangular box is 2 cm by 3 cm by 4 cm. What is its total surface area? Candidate answers: (A) 18 cm² (B) 24 cm² (C) 26 cm² (D) 52 cm². Solution: Opposite faces occur in pairs. Add 2(2×3+2×4+3×4)=52 cm². All six faces count because the box is closed. The final answer is 52 cm². The word closed matters: an open-top box would omit one face. Do not confuse lwh with the sum of face areas; surface material is measured in square units, not cubic units.
Check your reasoning
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Helpful tips
- A closed rectangular box has three pairs of congruent faces.
- The word closed matters: an open-top box would omit one face.
See the answer and explanation
94 cm²
Opposite faces occur in pairs. Add 2(3×4+3×5+4×5)=94 cm². All six faces count because the box is closed.
Take it further
Changed-input task: A closed rectangular box is 9 cm by 10 cm by 11 cm. What is its total surface area? Candidate answers: (A) 299 cm² (B) 598 cm² (C) 60 cm² (D) 990 cm². Show the mathematical reason, not only the selected answer. Expected result: 598 cm². Opposite faces occur in pairs. Add 2(9×10+9×11+10×11)=598 cm². All six faces count because the box is closed.
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