Mensuration, 3D and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Mensuration, 3D and Optimization short notes for CAT: 31 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Cuboid Formula Toolkit

Rectangular box with sides $a, b, c$

Volume:

V = ab c

Total surface area:

S = 2 (ab + b c + c a)

Sum of all edge lengths:

4 (a + b + c)

A cuboid has

12

edges: four of length

a

, four of length

b

, and four of length

c

Sphere Formula Toolkit

Sphere formulas

Surface area

4 π r^{2}

Volume

\frac{4}{3} π r^{3}

Sphere questions usually become easy once the radius is found.

Space Diagonal: 3D Pythagoras

Diagonal through the box

d^{2} = a^{2} + b^{2} + c^{2}

d = a^{2} + b^{2} + c^{2}

This is the key bridge between a box and a sphere.

Box Inscribed in Sphere: Diagonal Equals Diameter

Inscribed box inside sphere

Box space diagonal:

d = a^{2} + b^{2} + c^{2}

Sphere diameter:

2 r = d

Hence:

r = \frac{d}{2}

Mensuration, 3D and Optimization: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability MCQ

A closed rectangular box is inscribed in a sphere, so that all eight vertices of the box lie on the sphere. The three distinct face-diagonal lengths of the box are $13$ cm, $35$ cm and $210$ cm. The diameter of the sphere, in cm, is

A.
$\frac{7}{2}$
B.
$7$
C.
$98$
D.
$14$

Correct Answer:

Step-by-Step Solution

Key idea: this is a box-inscribed-in-a-sphere question. The space diagonal of the box is the diameter of the sphere. The twist is that the question gives the three face diagonals, not the edges.

Step 1: Let the box sides be $a, b, c$ . The three distinct face-diagonal squares are:

$a^{2} + b^{2}, b^{2} + c^{2}, c^{2} + a^{2} .$

Step 2: Square the given face diagonals and add them:

$(13)^{2} + (35)^{2} + (210)^{2} = 13 + 45 + 40 = 98.$

Step 3: Relate this sum to the space diagonal $d$ :

$(a^{2} + b^{2}) + (b^{2} + c^{2}) + (c^{2} + a^{2}) = 2 (a^{2} + b^{2} + c^{2}) = 2 d^{2} .$

So $2 d^{2} = 98$ , giving $d^{2} = 49$ and $d = 7$ .

Step 4: Since the box is inscribed in the sphere, the space diagonal is the sphere's diameter.

Answer: $7$ cm.

Common trap: using $98$ directly as $d^{2}$ forgets that each of $a^{2}, b^{2}, c^{2}$ appears twice in the sum of face-diagonal squares. Another trap is converting the space diagonal to radius instead of diameter.

Question 2 · Quantitative Ability NAT

A closed rectangular box has its length, breadth and height in the ratio $1 : 2 : 3$ . It is inscribed in a sphere of radius $14$ cm. What is the volume of the box, in cubic cm?

Correct Answer:

Step-by-Step Solution

Key idea: this is a box-inscribed-in-sphere question. The trigger is that all vertices of the rectangular box lie on the sphere, so the box's space diagonal becomes the sphere's diameter.

Step 1: Let the dimensions be $x, 2 x, 3 x$ .

Step 2: The sphere has radius $14$ , so its diameter is

$214 .$

Step 3: For a rectangular box, the space diagonal satisfies

$d^{2} = x^{2} + (2 x)^{2} + (3 x)^{2} .$

$d^{2} = x^{2} + 4 x^{2} + 9 x^{2} = 14 x^{2} .$

Step 4: Since the box is inscribed in the sphere,

$d = 214 .$

Therefore

$14 x^{2} = (214)^{2} = 4 \cdot 14 = 56.$

$x^{2} = 4 ⟹ x = 2.$

Step 5: The dimensions are $2, 4, 6$ . The volume is

$2 \cdot 4 \cdot 6 = 48.$

Answer: 48.

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Mensuration, 3D and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Mensuration, 3D and Optimization short notes for CAT: 31 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

A closed rectangular box is inscribed in a sphere, so that all eight vertices of the box lie on the sphere. The three distinct face-diagonal lengths of the box are 13 cm, 35 cm and 210 cm. The diameter of the sphere, in cm, is

Question 2

A closed rectangular box has its length, breadth and height in the ratio 1 : 2 : 3 . It is inscribed in a sphere of radius 14 cm. What is the volume of the box, in cubic cm?

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Mensuration, 3D and Optimization Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Cuboid Formula Toolkit

Rectangular box with sides a,b,c

Sphere Formula Toolkit

Sphere formulas

Space Diagonal: 3D Pythagoras

Diagonal through the box

Box Inscribed in Sphere: Diagonal Equals Diameter

Inscribed box inside sphere

Mensuration, 3D and Optimization: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

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Rectangular box with sides $a, b, c$