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The sum of length, breadth, and height of a cuboid is 12 cm and its diagonal is 8 cm. Find its total surface area.

80 sq cm

64 sq cm

144 sq cm

100 sq cm

Correct Answer

📐 MA • Math Direct FormulaMathematicsMensuration

Option A

80 sq cm

Quick Summary:

Use the algebraic identity $(l+b+h)^2 - (d^2) = \text{Total Surface Area}$ , where $d^2 = l^2+b^2+h^2$ . Simply calculate $12^2 - 8^2 = 144 - 64 = 80$ .

📐MAMath SolutionDirect Formula

📋 Given

Sum of dimensions (length, breadth, and height) = 12 cm, Diagonal of cuboid = 8 cm.

🔢 Formula Used

(l+b+h)2=l2+b2+h2+2(lb+bh+hl)(l+b+h)^2 = l^2+b^2+h^2 + 2(lb+bh+hl)(l+b+h)2=l2+b2+h2+2(lb+bh+hl)

⚡ Exam Hall Shortcut / Speed Trick

Use the algebraic identity $(l+b+h)^2 - (d^2) = \text{Total Surface Area}$ , where $d^2 = l^2+b^2+h^2$ . Simply calculate $12^2 - 8^2 = 144 - 64 = 80$ .

⚠️ Common Student Trap / Pitfall

Students often confuse the formula for the diagonal ( $d = \sqrt{l^2+b^2+h^2}$ ) with the sum of dimensions ( $l+b+h$ ) and try to solve for individual dimensions, which is not required.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define given variables

Let the dimensions of the cuboid be $l$ , $b$ , and $h$ . We are given the sum $l+b+h = 12$ and the diagonal $d = \sqrt{l^2+b^2+h^2} = 8$ .

l+b+h=12,l2+b2+h2=82=64l+b+h = 12, \quad l^2+b^2+h^2 = 8^2 = 64l+b+h=12,l2+b2+h2=82=64

Apply algebraic identity

The total surface area (TSA) of a cuboid is given by $2(lb+bh+hl)$ . We use the expansion of $(l+b+h)^2$ .

(l+b+h)2=l2+b2+h2+2(lb+bh+hl)(l+b+h)^2 = l^2+b^2+h^2 + 2(lb+bh+hl)(l+b+h)2=l2+b2+h2+2(lb+bh+hl)

Substitute and calculate

Substitute the known values into the equation: $12^2 = 64 + \text{TSA}$ . Solving for TSA gives $144 - 64 = 80$ .

TSA=144−64=80\text{TSA} = 144 - 64 = 80TSA=144−64=80

✅

A is correct because the total surface area calculated using the identity $(l+b+h)^2 - d^2$ results in 80 sq cm.

Core Concepts Used

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Algebraic identities Cuboid geometry Surface area calculation

💡 EXAM TIP

This identity-based approach is frequently tested in 'Algebra' sections of competitive exams, specifically under problems involving symmetric polynomials.

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