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

Mensuration, 3D and Optimization notes for CAT: 38 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Mensuration, 3D and Optimization

CAT QA Geometry

Mensuration, 3D and Optimization

A short chapter where formulas become fast decision tools.

chapter PYQs

①

📦 t1 — Three-Dimensional Mensuration

You master cuboids, spheres, surface area, volume, space diagonal, and 3D enclosure logic.

1 own-course PYQ Selected topic Formula + algebra

②

📐 t2 — Area Optimization in Plane Figures

You learn how geometric restrictions force a maximum area.

1 own-course PYQ

③

💰 t3 — Perimeter and Cost Optimization

You convert side lengths and area constraints into minimum-cost decisions.

1 own-course PYQ

End goal: identify whether the question is asking for a 3D measure, a maximum area, or a minimum cost, then apply the shortest formula path.

Topic Hero: Three-Dimensional Mensuration

Geometry → Mensuration → t1

Three-Dimensional Mensuration

One-line hook: in 3D, the hidden diagonal often controls the whole solid.

What you'll learn here

Cuboid surface area, volume, and total edge length.
Sphere radius, surface area, and volume basics.
Space diagonal of a rectangular box.
Why an inscribed box has diagonal equal to sphere diameter.
How CAT combines formulas using algebraic identities.

3D Mensuration: Length, Area, Volume

Three things you measure in 3D

📏

Length

Edge, radius, height, or diagonal.

🧻

Surface area

Outer wrapping of the solid.

📦

Volume

Space occupied by the solid.

CAT habit: decide first whether the question asks for edge length, surface area, or volume.

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

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.

More notes in this unit

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

Mensuration, 3D and Optimization notes for CAT: 38 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 Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Chapter Roadmap: Mensuration, 3D and Optimization

Mensuration, 3D and Optimization

📦 t1 — Three-Dimensional Mensuration

📐 t2 — Area Optimization in Plane Figures

💰 t3 — Perimeter and Cost Optimization

Topic Hero: Three-Dimensional Mensuration

Three-Dimensional Mensuration

What you'll learn here

3D Mensuration: Length, Area, Volume

Three things you measure in 3D

Length

Surface area

Volume

Cuboid Formula Toolkit

Rectangular box with sides a,b,c

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

Step-by-Step Solution

Step-by-Step Solution

More notes in this unit

A question from this chapter

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Rectangular box with sides $a, b, c$