Study Notes on Mensuration (2D)

Rectangle

Let d₁ and d₂ are diagonals of the given rectangle ABCD.

then, both diagonals are equal but not perpendicular to each other.

 Area of rectangle = length x breadth and perimeter = 2(length+breadth)

Path outside the rectangle

Suppose there is a park having length l and breadth b. There is a road of width x outside of it.

Then, Area of path = 2x (l + b + 2x)

Path inside the rectangle

Suppose there is a park having length l and breadth b. There is a road of width x inside of it.

Then, Area of path = 2x (l + b – 2x)

When there is a road along both the length and breadth of the park.

Then, Remaining area of Rectangle (shaded region) = (l–x) (b-x)
Area of the path = lx + bx – x²

Circle: Given a circle of radius ‘r’

We recommend you learn this table as it will save your time in calculating these all.

If radius is ‘r’, then perimeter = 2πr and Area = πr²

Length of Rope
Let ‘d’ is the diameter of the pulley and ‘r’ is the radius, then d = 2r. All pulleys are similar.

Length of rope = 2d + 2pr

Length of rope = 3d + 2pr

Length of rope = 4d + 2pr

Note: Trick to remember these formulas: number of pulleys x diameter + Perimeter of one pulley

Sector

In this circle, ‘r’ is the radius, θ is the angle made by the arc of length ‘l’

Segment

Area of the minor segment 

Area of a major segment 

Area of the shaded portion

Inradius and Circumradius of Square:

There is a square of side ‘a’; ‘r’ is the inradius and ‘R’ is the circumradius.

Triangle:

Let ABC is a triangle and M₁, M₂ and M₃ are medians of the given triangle.

Then, 

Inradius of the triangle:

Given, ABC is a triangle and a, b and c are the sides of the given triangle. Let ‘r’ is the inradius of the triangle.

Circumradius of the triangle:

Given, ABC is a triangle and a, b and c are the sides of the given triangle. Let ‘R’ is the circumradius of the triangle.

Right angle triangle

Given ‘a’ is the base, ‘b’ is the perpendicular and ‘c’ is the hypotenuse of triangle ABC.

Equilateral triangle:

Where h is the height of the triangle, 

Hence, we can say that height of the equilateral triangle is equal to the sum of the side perpendicular to the triangle.

# Isosceles triangle

# Regular Polygon

Let, n = no. of sides of regular polygon and a = length of side of regular polygon

# Internal angle of regular polygon = 

# Sum of internal angle of regular polygon  

# Angle made by centre = 

#Area of Regular polygon 

 or  

# External angle of regular polygon 
# sum of all external angle = 360º

# For Regular Hexagon

# Cyclic Quadrilateral

 Parallelogram

Let a and b are the sides, h is the height and d₁ and d₂ are the diagonals of parallelogram

then, 

Area of parallelogram = (i) Base × height

(ii) 

(iii) 

Imp. Relation 

Imp. Note: In rectangle, parallelogram, square and Rhombus diagonals bisect other.

Rhombus

In Rhombus, diagonals are not equal to each other but they bisect each other at 90 degree. 

Area = Base × height = a x h

Or Area 

Trapezium

Case 1: If AD = BC, then DM = CN

Quadrilateral