Mensuration - All Formulas & Shortcuts

Complete formula reference for 2D and 3D shapes with quick calculation tricks

📘 2D Shapes - Basic Formulas

1. Square

Side = a

Perimeter = 4a Area = a² Diagonal = a√2

Example: Side = 10 cm

Perimeter = 40 cm Area = 100 cm² Diagonal = 10√2 = 14.14 cm

2. Rectangle

Length = l, Breadth = b

Perimeter = 2(l + b) Area = l × b Diagonal = √(l² + b²)

Example: l=12, b=5

Perimeter = 2(17) = 34 Area = 60 Diagonal = √(144+25) = √169 = 13

3. Circle

Radius = r, Diameter = d = 2r

Circumference = 2πr = πd Area = πr²

Use π = 22/7 or 3.14

Example: r = 7 cm

Circumference = 2 × 22/7 × 7 = 44 cm Area = 22/7 × 49 = 154 cm²

4. Triangle

General triangle (base = b, height = h):

Area = (1/2) × b × h

Equilateral triangle (side = a):

Area = (√3/4) × a² Height = (√3/2) × a Perimeter = 3a

Scalene triangle (sides a, b, c):

s = (a+b+c)/2 (semi-perimeter) Area = √[s(s-a)(s-b)(s-c)] (Heron’s formula)

Right triangle (base=b, height=h, hypotenuse=H):

H² = b² + h² (Pythagoras) Area = (1/2) × b × h

5. Parallelogram

Base = b, Height = h

Area = b × h Perimeter = 2(a + b) where a, b are sides

6. Rhombus

Side = a, Diagonals = d₁, d₂

Area = (1/2) × d₁ × d₂ Perimeter = 4a

Relation:

4a² = d₁² + d₂²

7. Trapezium

Parallel sides = a, b; Height = h

Area = (1/2) × (a + b) × h

Example: a=8, b=6, h=5

Area = (1/2) × 14 × 5 = 35

🔥 3D Shapes - Volume & Surface Area

8. Cube

Side = a

Volume = a³ Total Surface Area (TSA) = 6a² Lateral Surface Area (LSA) = 4a² Diagonal = a√3

Example: a = 5

Volume = 125 TSA = 150 Diagonal = 5√3 = 8.66

9. Cuboid (Rectangular Box)

Length = l, Breadth = b, Height = h

Volume = l × b × h TSA = 2(lb + bh + hl) LSA = 2h(l + b) Diagonal = √(l² + b² + h²)

Example: l=10, b=5, h=4

Volume = 200 TSA = 2(50+20+40) = 220

10. Cylinder

Radius = r, Height = h

Volume = πr²h Curved Surface Area (CSA) = 2πrh TSA = 2πr(r + h)

Example: r=7, h=10

Volume = 22/7 × 49 × 10 = 1540 CSA = 2 × 22/7 × 7 × 10 = 440 TSA = 2 × 22/7 × 7 × 17 = 748

11. Cone

Radius = r, Height = h, Slant height = l

Slant height: l = √(r² + h²) Volume = (1/3)πr²h CSA = πrl TSA = πr(r + l)

Example: r=7, h=24

l = √(49+576) = √625 = 25 Volume = (1/3) × 22/7 × 49 × 24 = 1232 CSA = 22/7 × 7 × 25 = 550

12. Sphere

Radius = r

Volume = (4/3)πr³ Surface Area = 4πr²

Example: r=7

Volume = (4/3) × 22/7 × 343 = 1437.33 Surface Area = 4 × 22/7 × 49 = 616

13. Hemisphere

Radius = r

Volume = (2/3)πr³ CSA = 2πr² TSA = 3πr²

Example: r=7

Volume = (2/3) × 22/7 × 343 = 718.67 CSA = 2 × 22/7 × 49 = 308 TSA = 3 × 22/7 × 49 = 462

⚡ Quick Shortcuts

14. Area Relationships

Circle vs Square (same perimeter):

If perimeter is P: Circle area > Square area Circle area = P²/(4π) Square area = P²/16

Example: Perimeter = 44 cm

Circle: 44²/(4π) = 1936/12.56 = 154 cm² Square: 44²/16 = 121 cm²

15. Volume Scale Factor

If all dimensions scaled by factor k:

New Volume = k³ × Original Volume New Surface Area = k² × Original SA

Example: Cube side doubled (k=2)

Volume becomes 2³ = 8 times Surface area becomes 2² = 4 times

16. Hollow Cylinder

Outer radius = R, Inner radius = r, Height = h

Volume of material = π(R² - r²)h CSA outer = 2πRh CSA inner = 2πrh Total CSA = 2π(R + r)h TSA = 2π(R² - r²) + 2π(R + r)h

17. Frustum of Cone

Top radius = r, Bottom radius = R, Height = h

Volume = (1/3)πh(R² + r² + Rr) Slant height = √[h² + (R-r)²] CSA = π(R + r) × slant height

💡 Exam Patterns

Pattern 1: Ratio of Areas

When dimensions in ratio:

If sides in ratio a:b, areas in ratio a²:b²

Example: Squares with sides 3:4

Area ratio = 9:16

Pattern 2: Path/Border Problems

Path around rectangle (width = w):

Area of path = 2w(l + b + 2w)

Path inside rectangle:

Area of path = 2w(l + b - 2w)

Pattern 3: Four Walls of Room

Room: l × b × h

Area of 4 walls = 2h(l + b)

Example: 10m × 8m × 6m

4 walls = 2 × 6 × 18 = 216 m²

Pattern 4: Melting/Recasting

Volume remains same:

Example: Sphere melted into cylinder

(4/3)πr³ = πR²h Solve for unknown

📊 Quick Reference Table

2D Shapes

3D Shapes

Shape	Volume	Surface Area
Cube (a)	a³	6a²
Cuboid	lbh	2(lb+bh+hl)
Cylinder	πr²h	2πr(r+h)
Cone	⅓πr²h	πr(r+l)
Sphere	⅘πr³	4πr²

💎 Mental Math Tricks

Trick 1: π Approximations

Quick calculations:

π = 22/7 ≈ 3.14 π² ≈ 10 √π ≈ 1.77

Trick 2: Common Squares

11² = 121 12² = 144 13² = 169 14² = 196 15² = 225

Trick 3: Pythagorean Triples

Common right triangles:

3-4-5 5-12-13 8-15-17 7-24-25

🔍 Common Mistakes

❌ Using diameter instead of radius ❌ Forgetting (1/3) factor in cone/pyramid ❌ Confusing TSA with CSA ❌ Not converting units (m to cm) ✅ Always check which surface area is asked ✅ Remember: Volume has cubic units, Area has square units ✅ Sketch diagram when confused ✅ Double-check formula before calculation

🔗 Related Resources

Practice Questions:

• Mensuration Practice Questions

Theory:

• Mensuration Theory & Concepts

Related Topics:

• Percentage
• Ratio & Proportion

Study Resources:

• All-in-One Formula Sheet

Memorize 2D formulas first, then 3D becomes easy! ⚡

Remember: Volume always has ×h, Surface area has +h! 🚀