The areas of three adjacent faces of a solid cuboid are 260 cm... | Mathematics MCQ | Examoogle

MathematicsMensuration

The areas of three adjacent faces of a solid cuboid are 260 cm 2, 104 cm 2 and 10 cm 2. What is the volume (in cm 3 ) of the cuboid?

627

520

524

782

Correct Answer

📐 MA • Math Concept & PrincipleMathematicsMensuration

Option B

520

Quick Summary:

The volume of a cuboid is the product of its length ( $l$ ), breadth ( $b$ ), and height ( $h$ ). Given the areas of the adjacent faces are $A_{1} = lb$ , $A_{2} = bh$ , and $A_{3} = hl$ , the volume $V$ is equal to the square root of the product of these three areas: $V = \sqrt{A_{1} \cdot A_{2} \cdot A_{3}}$ .

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💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of this as calculating the total space inside a box by multiplying the dimensions of its three visible side views, where the interaction of the overlapping edges balances out to represent the 3D interior capacity.

🧠 Memory Mnemonic / Shortcut Aid

Area Areas Area Product Rooted is Volume: $V = \sqrt{A_{1}A_{2}A_{3}}$ .

🔢 Key Formulas

$V = \sqrt{(lb)(bh)(hl)} = \sqrt{A_{1}A_{2}A_{3}}$ — Volume calculation from face areas

$A = l \cdot w$ — Area of a rectangle

⚙️ Working Principle

The principle relies on the property of geometric products. If we multiply the three areas together: $(lb) \cdot (bh) \cdot (hl) = l^{2}b^{2}h^{2}$ . Taking the square root of this product yields $\sqrt{l^{2}b^{2}h^{2}} = lbh$ , which is the definition of volume for a rectangular parallelepiped.

📌 Key Points

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A cuboid has three pairs of congruent faces.
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Adjacent faces share an edge; the product of three distinct face areas is the square of the volume.
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This method is derived from the geometric mean principle applied to spatial dimensions.

✅ Advantages

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Allows volume calculation without knowing individual dimensions $l, b, h$ .

❌ Disadvantages / Limitations

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Only applicable to rectangular cuboids with right-angled corners.

🛠️ Applications / Uses

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Packaging engineering
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Architecture and spatial volume estimation

📄 Additional Information

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Calculation: $V = \sqrt{260 \times 104 \times 10} = \sqrt{270400} = 520$ .
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Option B is the correct result of the square root calculation.

📊 Diagram / Illustration

✅

B is correct — The volume is determined by the square root of the product of the areas of its three adjacent faces, calculated as $\sqrt{260 \times 104 \times 10} = 520$ cm $^3$ .

Core Concepts Used

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Geometric Volume Spatial Geometry Square Roots

💡 EXAM TIP

Whenever you encounter a geometry problem involving adjacent faces, always look for the relationship between the product of areas and the volume; it saves time by avoiding solving for individual variables.

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