The areas of three adjacent faces of a solid cuboid are 297 cm², 33 cm² and 196 cm². What is the volume (in cm³) of the cuboid?

Appeared in: DSSSB 12 August 2024

Explanation

The volume (V) of a cuboid is related to the areas of its three adjacent faces (A₁, A₂, A₃) by the formula: V = √(A₁ × A₂ × A₃).
The given areas are 297 cm², 33 cm², and 196 cm².
First, calculate the product of these areas: 297 × 33 × 196 = 1,920,996.
Next, find the square root of this product to determine the volume: √1,920,996 = 1386.
Therefore, the volume of the cuboid is 1386 cm³.

Option A: This is an incorrect calculation. If the volume were 1310 cm³, the product of the face areas would have to be 1310², which is 1,716,100, not 1,920,996.
Option B: This is an incorrect calculation. If the volume were 1307 cm³, the product of the face areas would have to be 1307², which is 1,708,249, not 1,920,996.
Option C: This is an incorrect calculation. If the volume were 1173 cm³, the product of the face areas would have to be 1173², which is 1,375,929, not 1,920,996.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Volume of a Cuboid from Adjacent Face Areas as background academic context rather than a clinical decision trigger.
This question tests foundational knowledge of geometry (mensuration), a common topic in the quantitative aptitude section of nursing and other competitive entrance exams.
Mastering such formulas allows for quick and accurate problem-solving under time pressure, a critical skill for succeeding in these exams.
What if the question provided the volume and two face areas? If the volume was 1386 cm³ and two face areas were 297 cm² and 33 cm², you would find the third area by rearranging the formula: Area₃ = (Volume²) / (Area₁ × Area₂), which would be 1386² / (297 × 33) = 196 cm².

How to Approach the Question

First, identify the given information: the areas of three adjacent faces of a cuboid (297 cm², 33 cm², and 196 cm²).
Recall the relationship between the volume of a cuboid and the areas of its adjacent faces. Let the dimensions be l, b, and h. The areas are A₁=lb, A₂=bh, A₃=hl.
Understand that the product of these areas is A₁×A₂×A₃ = (lb)(bh)(hl) = l²b²h² = (lbh)² = Volume².
From this, derive the formula for the volume: Volume = √(A₁ × A₂ × A₃).
Substitute the given values into the formula: Volume = √(297 × 33 × 196).
Calculate the product of the areas: 297 × 33 × 196 = 1,920,996.

Concept Tested & Keywords

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