Revison Complete Mensuration Formula : Area, Volume & Surface Area

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Complete Mensuration Formula PDF

Mensuration is the branch of mathematics used to calculate the perimeter, area, surface area and volume of geometrical figures.

This complete formula sheet covers important 2D and 3D geometrical shapes useful for competitive examinations.

Important 2D Shapes

  • Circle
  • Semicircle
  • Square
  • Rectangle
  • Parallelogram
  • Rhombus
  • Trapezium
  • Triangle
  • Quadrilateral

Important 3D Shapes

  • Cube
  • Cuboid
  • Cylinder
  • Cone
  • Frustum
  • Sphere
  • Hemisphere
  • Prism
  • Pyramid
Exam Tip: Before applying a formula, identify the shape and determine whether the question asks for perimeter, area, surface area or volume.

Perimeter Formulas

The perimeter is the total length of the boundary of a two-dimensional figure.

Shape Perimeter Formula
Square 4a
Rectangle 2(l + b)
Triangle a + b + c
Quadrilateral a + b + c + d
Parallelogram 2(a + b)
Rhombus 4a
Trapezium a + b + c + d
Circle 2πr
Semicircle πr + 2r
Example:

A rectangle has length 12 cm and breadth 8 cm. Find its perimeter.

P = 2(l + b)
P = 2(12 + 8) = 40 cm
Answer: 40 cm

Area Formulas

Area represents the amount of surface covered by a two-dimensional figure.

Shape Area
Square a²
Rectangle l × b
Triangle ½ × Base × Height
Equilateral Triangle √3/4 × a²
Parallelogram Base × Height
Rhombus ½ × d₁ × d₂
Trapezium ½ × (a+b) × h
Circle πr²
Semicircle ½πr²
Example: Rectangle Area = 15 × 8 = 120 cm²

Unit: Area is expressed in square units such as cm², m² and km².

Circumference Formulas

The circumference is the distance around a circle. It is also called the perimeter of a circle.

Circumference = 2πr
Diameter = 2r
Circumference = πd

Where:

  • r = Radius
  • d = Diameter
  • π = 22/7 or 3.14
Example:

Find the circumference of a circle with radius 7 cm.

C = 2 × 22/7 × 7
C = 44 cm

Surface Area Formulas

Surface area is the total area covering the outer surface of a three-dimensional solid.

Solid Important Surface Area
Cube TSA = 6a²
Cuboid TSA = 2(lb + bh + hl)
Cylinder TSA = 2πr(r+h)
Cone TSA = πr(r+l)
Sphere Surface Area = 4πr²
Hemisphere TSA = 3πr²
Prism TSA = LSA + 2 × Base Area
Pyramid TSA = LSA + Base Area

Surface area is measured in square units such as cm² and m².

Curved Surface Area (CSA)

Curved Surface Area refers to the area of the curved portion of a three-dimensional solid, excluding its flat circular or polygonal bases.

Cylinder

CSA = 2πrh

Cone

CSA = πrl

Hemisphere

CSA = 2πr²

Where r is radius, h is perpendicular height and l is slant height.

Example: Cylinder with r = 7 cm and h = 10 cm
CSA = 2 × 22/7 × 7 × 10
CSA = 440 cm²

Lateral Surface Area (LSA)

Lateral Surface Area is the area of the side faces of a solid, excluding its bases.

Cube

LSA = 4a²

Cuboid

LSA = 2(l+b)h

Prism

LSA = Perimeter of Base × Height

Pyramid

LSA = ½ × Perimeter of Base × Slant Height

Example: If a prism has base perimeter 30 cm and height 10 cm:

LSA = 30 × 10 = 300 cm²

Total Surface Area (TSA)

Total Surface Area is the complete outer surface area of a three-dimensional solid.

Cube: TSA = 6a²
Cuboid: TSA = 2(lb + bh + hl)
Cylinder: TSA = 2πr(r+h)
Cone: TSA = πr(r+l)
Sphere: TSA = 4πr²
Hemisphere: TSA = 3πr²
Prism: TSA = LSA + 2 × Base Area
Pyramid: TSA = LSA + Base Area
Example: Cube with side 5 cm
TSA = 6 × 5² = 150 cm²

Volume Formulas

Volume measures the amount of space occupied by a three-dimensional solid.

Solid Volume
Cube a³
Cuboid l × b × h
Cylinder πr²h
Cone ⅓πr²h
Frustum ⅓π(R² + r² + Rr)h
Sphere ⁴⁄₃πr³
Hemisphere ²⁄₃πr³
Prism Base Area × Height
Pyramid ⅓ × Base Area × Height

Volume is expressed in cubic units such as cm³, m³ and km³.

2D Shapes – Area & Perimeter

Two-dimensional figures have length and breadth but no volume. Important 2D mensuration calculations include area and perimeter.

Shape Area Perimeter
Square a² 4a
Rectangle l × b 2(l+b)
Triangle ½bh a+b+c
Parallelogram bh 2(a+b)
Rhombus ½d₁d₂ 4a
Trapezium ½(a+b)h a+b+c+d
Circle πr² 2πr
Semicircle ½πr² πr+2r

3D Shapes – Surface Area & Volume

Solid Surface Area Volume
Cube 6a² a³
Cuboid 2(lb+bh+hl) lbh
Cylinder 2πr(r+h) πr²h
Cone πr(r+l) ⅓πr²h
Sphere 4πr² ⁴⁄₃πr³
Hemisphere 3πr² ²⁄₃πr³
Prism LSA+2B Bh
Pyramid LSA+B ⅓Bh

Triangle Area Formulas

General Triangle

Area = ½ × Base × Height
Area = √[s(s-a)(s-b)(s-c)]
Area = ½ab sinθ

Where: s = (a+b+c)/2 is the semiperimeter.

Equilateral Triangle

Area = √3/4 × a²

Isosceles Triangle

Area = b/4 × √(4a²-b²)
Area = ½a²sinα

Right-Angled Triangle

Area = ½ × Base × Height
Example: Base = 8 cm, Height = 6 cm
Area = ½ × 8 × 6 = 24 cm²

Circle Area & Circumference

r O Circle
Diameter = 2r
Circumference = 2πr
Area = πr²

Take π = 22/7 or 3.14 as required by the question.

Example: r = 7 cm
Area = 22/7 × 49
Area = 154 cm²

Square & Rectangle Formulas

Square

Area = a²
Perimeter = 4a
Diagonal = √2a
Inradius = a/2
Circumradius = a/√2

Rectangle

Area = l × b
Perimeter = 2(l+b)
Diagonal = √(l²+b²)
Area of path inside rectangle = 2x(l+b-2x)
Area of path outside rectangle = 2x(l+b+2x)
Area of path in middle = x(l+b-x)
Example: Square side = 10 cm
Area = 10² = 100 cm²

Quadrilateral Formulas

A quadrilateral has 4 sides, 4 vertices and 4 angles.

Sum of Interior Angles = 360°
Perimeter = a+b+c+d
Area = ½ × AC × (BN+DM)

Parallelogram

Area = Base × Height
Perimeter = 2(a+b)

Rhombus

Area = ½d₁d₂
Perimeter = 4a

Trapezium

Area = ½(a+b)h
Perimeter = a+b+c+d

Quadrilateral Formulas

A quadrilateral has 4 sides, 4 vertices and 4 angles.

Sum of Interior Angles = 360°
Perimeter = a+b+c+d
Area = ½ × AC × (BN+DM)

Parallelogram

Area = Base × Height
Perimeter = 2(a+b)

Rhombus

Area = ½d₁d₂
Perimeter = 4a

Trapezium

Area = ½(a+b)h
Perimeter = a+b+c+d

Cube & Cuboid Formulas

Cube

Volume = a³
LSA = 4a²
TSA = 6a²
Diagonal = √3a

Cuboid

Volume = lbh
LSA = 2(l+b)h
TSA = 2(lb+bh+hl)
Diagonal = √(l²+b²+h²)
Example: Cube side = 5 cm
Volume = 5³ = 125 cm³
TSA = 6 × 25 = 150 cm²

Cylinder & Cone Formulas

Cylinder

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

Cone

Slant Height = √(r²+h²)
Volume = ⅓πr²h
CSA = πrl
TSA = πr(r+l)
Example: Cone with r = 7 cm and h = 24 cm
l = √(49+576) = 25 cm

Sphere & Hemisphere Formulas

Sphere

Surface Area = 4πr²
Volume = ⁴⁄₃πr³

Hemisphere

CSA = 2πr²
TSA = 3πr²
Volume = ²⁄₃πr³
Example: Sphere radius = 7 cm
Surface Area = 4 × 22/7 × 49
= 616 cm²

Prism & Pyramid Formulas

Prism

LSA = Perimeter of Base × Height
TSA = LSA + 2 × Area of Base
Volume = Area of Base × Height

Pyramid

LSA = ½ × Perimeter of Base × Slant Height
TSA = LSA + Base Area
Volume = ⅓ × Base Area × Height
Example: Pyramid with base area 60 cm² and height 9 cm
Volume = ⅓ × 60 × 9
= 180 cm³

Mensuration Formula Chart

Shape Area / Surface Area Volume
Square a² —
Rectangle lb —
Circle πr² —
Cube 6a² a³
Cuboid 2(lb+bh+hl) lbh
Cylinder 2πr(r+h) πr²h
Cone πr(r+l) ⅓πr²h
Sphere 4πr² ⁴⁄₃πr³
Hemisphere 3πr² ²⁄₃πr³
Prism LSA+2B Bh
Pyramid LSA+B ⅓Bh

Mensuration Formula Table

Figure Formula Quantity
Square a² Area
Square 4a Perimeter
Rectangle lb Area
Rectangle 2(l+b) Perimeter
Triangle ½bh Area
Circle πr² Area
Circle 2πr Circumference
Cube a³ Volume
Cylinder πr²h Volume
Cone ⅓πr²h Volume
Sphere ⁴⁄₃πr³ Volume
Hemisphere ²⁄₃πr³ Volume

Important Units & Conversions

Length

Conversion Value
1 metre 100 centimetres
1 kilometre 1000 metres
1 centimetre 10 millimetres

Area

Conversion Value
1 m² 10,000 cm²
1 cm² 100 mm²
1 km² 1,000,000 m²

Volume

Conversion Value
1 m³ 1,000,000 cm³
1 cm³ 1000 mm³
1 litre 1000 cm³
1 millilitre 1 cm³
Remember: Length uses units such as cm and m; area uses cm² and m²; volume uses cm³ and m³.

Complete Mensuration Formula Revision Sheet

2D Figures

Square Area = a²
Square Perimeter = 4a
Rectangle Area = l × b
Rectangle Perimeter = 2(l+b)
Triangle Area = ½bh
Triangle Perimeter = a+b+c
Circle Area = πr²
Circle Circumference = 2πr
Semicircle Area = ½πr²
Semicircle Perimeter = πr+2r
Parallelogram Area = bh
Rhombus Area = ½d₁d₂
Trapezium Area = ½(a+b)h

3D Figures

Cube Volume = a³
Cube TSA = 6a²
Cuboid Volume = lbh
Cuboid TSA = 2(lb+bh+hl)
Cylinder Volume = πr²h
Cylinder CSA = 2πrh
Cylinder TSA = 2πr(r+h)
Cone Volume = ⅓πr²h
Cone CSA = πrl
Cone TSA = πr(r+l)
Sphere Surface Area = 4πr²
Sphere Volume = ⁴⁄₃πr³
Hemisphere CSA = 2πr²
Hemisphere TSA = 3πr²
Hemisphere Volume = ²⁄₃πr³
Prism Volume = Base Area × Height
Prism LSA = Base Perimeter × Height
Prism TSA = LSA + 2 × Base Area
Pyramid Volume = ⅓ × Base Area × Height
Pyramid LSA = ½ × Base Perimeter × Slant Height
Pyramid TSA = LSA + Base Area

Quick Exam Reminder

Area → square units

Perimeter → linear units

Surface Area → square units

Volume → cubic units

π → 22/7 or 3.14