GEOMETRY & MENSURATION VAULT ✦ TRIANGLES & CIRCLES ✦ 3D VOLUMES ✦ NO MATH ANXIETY ✦ GEOMETRY & MENSURATION VAULT ✦
Geometry & Space

Geometry & Mensuration

Triangles, Circles, Tangent Theorems, Inradius & Circumradius, Quadrilaterals, and 2D/3D Mensuration volume formulas.

The Bodhi Vault / Quant Vault / Geometry & Mensuration
DEFINITION

One-Line Definition

Geometry studies the properties of points, lines, angles, triangles, and circles. Mensuration measures 2D perimeter/area and 3D surface area/volume.

Pythagoras: a² + b² = c² | Triangle Angle Sum = 180°
CORE INTUITION 📐

Blueprint & Container Model

Understanding 2D and 3D spatial properties:

  • • 2D Shapes: Floorplans (Triangles, Circles, Quadrilaterals) → Measure Area
  • • 3D Solids: Water Tanks (Cylinders, Cones, Spheres) → Measure Volume & Surface Area
  • • Melting a solid preserves VOLUME, not surface area!
Inradius (r) touches inside edges; Circumradius (R) passes through all vertices!
💡 WHY THIS CONCEPT MATTERS IN MBA EXAMS

Geometry accounts for 20-25% of CAT Quant questions! Knowing key theorems and volume formulas guarantees fast scoring:

CAT Quant Heavyweight
MBA CET 2D Mensuration
Right Triangle Inradius
Tangent Secant Theorem
Apollonius Theorem
3D Sphere & Cone Volume
Coordinate Geometry
Polygon Diagonals
📐 CORE FORMULAS & 2D/3D MENSURATION MATRIX

Triangle Area Formulas

Area = ½ × base × height

Heron's Area = √[s(s-a)(s-b)(s-c)]

Equilateral Area = (√3 / 4) × a²

3D Mensuration Volumes

Cylinder Volume = π r² h

Cone Volume = ⅓ π r² h (l = √(r² + h²))

Sphere Volume = (4/3) π r³ (SA = 4πr²)

SHORTCUT RULES & KEY FORMULAS

RIGHT TRIANGLE SHORTCUT

Inradius & Circumradius

Inradius r = (a + b - c) / 2 | Circumradius R = c / 2

For any right triangle with perpendicular legs a, b and hypotenuse c.

⚡ Mental Shortcut: For a 6-8-10 right triangle, Inradius r = (6 + 8 - 10)/2 = 2 cm, Circumradius R = 10/2 = 5 cm.
CIRCLE THEOREM

Tangent-Secant Theorem

PT² = PA × PB

From an external point P, PT is a tangent segment and PAB is a secant line passing through the circle.

⚡ Mental Shortcut: If PT = 12 cm and PA = 8 cm, then PB = 144 / 8 = 18 cm → Chord AB = 18 - 8 = 10 cm.

SOLVED EXAMPLES (LEVEL 0 TO HARD)

Example 1 (Easy / Level 0 Inradius)

Q: Find the inradius of a right triangle with sides 6 cm, 8 cm, and 10 cm.

Step 1: Identify legs a = 6, b = 8 and hypotenuse c = 10.

Step 2: Apply shortcut formula: r = (a + b - c) / 2 = (6 + 8 - 10) / 2 = 4 / 2 = 2 cm.

Answer = 2 cm
Example 2 (Medium / 3D Recasting Volume)

Q: A metallic sphere of radius 6 cm is melted and recast into a cone of base radius 6 cm. Find the height of the cone.

Step 1: Volume of Sphere = (4/3) π r³ = (4/3) π (6³).

Step 2: Volume of Cone = (1/3) π r² h = (1/3) π (6²) h.

Step 3: Equate volumes: (4/3) π (216) = (1/3) π (36) h → 4 × 216 = 36 h → h = 24 cm.

Answer = 24 cm
Example 3 (Hard / Tangent Secant Theorem)

Q: From an external point P, a tangent PT of length 12 cm is drawn to a circle. A secant PAB passes through the circle such that PA = 8 cm. Find chord length AB.

Step 1: Apply Tangent-Secant Theorem: PT² = PA × PB.

Step 2: 12² = 8 × PB → 144 = 8 × PB → PB = 18 cm.

Step 3: Chord AB = PB - PA = 18 - 8 = 10 cm.

Answer = 10 cm
⚠️ COMMON MISTAKES TO AVOID IN CAT & CET

Mistake 1

Confusing Inradius & Circumradius

Confusing inradius r = (a+b-c)/2 with circumradius R = c/2 in right-angled triangles.

Mistake 2

Forgetting Slant Height in Cone

Using vertical height h instead of slant height l = √(r² + h²) when calculating curved surface area πrl of a cone.

Mistake 3

Surface Area Recasting Trap

Assuming surface area is conserved when melting 3D solids. Only VOLUME remains constant during melting/recasting!

📝 PRACTICE QUESTIONS
Basic

1. Find the area of an equilateral triangle with side length 6 cm.

Answer: Area = (√3 / 4) × 6² = (√3 / 4) × 36 = 9√3 cm² (approx 15.59 cm²).

Moderate

2. Find the number of diagonals in a regular decagon (10 sides).

Answer: Diagonals = n(n - 3) / 2 = 10(7) / 2 = 35 diagonals.

Advanced

3. A right circular cylinder has height 14 cm and base radius 3 cm. Find its total surface area.

Answer: TSA = 2πr(r + h) = 2 × (22/7) × 3 × (3 + 14) = (132/7) × 17 = 320.57 cm².

FREQUENTLY ASKED QUESTIONS

❓ How do you find the inradius of a right-angled triangle in seconds?

For a right-angled triangle with legs a, b and hypotenuse c, Inradius r = (a + b - c) / 2. Circumradius R = c / 2.

❓ What is the Tangent-Secant theorem formula?

If a tangent PT and a secant PAB are drawn from an external point P to a circle, then PT² = PA × PB.

❓ How do you calculate the number of diagonals in an n-sided polygon?

The number of diagonals in any polygon with n sides is given by n(n - 3) / 2.