Mensuration Formula

Mensuration Formulas, Tricks with Examples

Mensuration is the branch of mathematics that studies the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc. Here, the concepts of mensuration are explained and all the important mensuration formulas and properties of different geometric shapes and figures are covered.

It is one of the most important chapters covered in high school Mathematics. It has immense practical applications in our day-to-day life. It is, for this reason, advanced concepts related to mensuration are covered in higher grades. It is also an important and scoring topic for competitive exams, like the Olympiads and NTSE. Mensuration problems are asked in various government job exams as well, like SSC, Banking, Insurance, etc. It is, therefore, important for everyone to understand and memorize the various mensuration formulas of all geometrical figures.

Mensuration Formulas: Area Formulas & Perimeter Formulas

Before getting into the list of Mensuration formulas, i.e. area formulas and perimeter formulas of all major geometric shapes, let’s have a look at the definition of ‘area’ and ‘perimeter’.

What Is Area Of A Geometric Figure?

The area of a closed geometric figure is basically the size or extent of the two-dimensional surface of the figure.

What Is Perimeter Of A Geometric Figure?

The perimeter of a closed geometric figure is the total length of the boundary of the figure.

Let us now go through the area and perimeter formulas of common f=geometric figures.

Area Formulas Of Common Geometric Figures

The table below shows the area formulas of common geometrical figures:

Geometrical Shapes Area Formulas Variables
Area of Square Area = a x a = a2 a = Length of each side of the square
Area of Rectangle Area = l x b l = Length of the rectangle
b = Breadth of the rectangle
Area of Triangle Area = (b x h)/2 b = Length of the base of the triangle
h = Height of the triangle
Area of Circle πr2 π = 22/7 = 3.14159 (approx)
r = Radius of the circle
Area of Parallelogram Area = b x h b = Base of the parallelogram
h = Height of the parallelogram
Area of Trapezium Area = {(a + b) x h}/2 a + b= Sum of the lengths of the two parallel sides of the trapezoid, i.e. Base 1 and Base 2
h = Perpendicular distance between the two parallel sides
Area of Rhombus Area = (p x q)/2 p = Length of 1st diagonal
q = Length of second diagonal
pq = Product of the two diagonals
Area of Ellipse Area = πab a = Major radius
b = Minor radius

Perimeter Formulas Of Common Geometrical Figures

The perimeter formulas of common geometrical figures are as under:

Geometrical Shapes Perimeter Formulas Variables
Perimeter of Square Perimeter = 4a a = Length of each side of the square
Perimeter of Rectangle Perimeter = 2 (l + b) l = Length of the rectangle
b = Breadth of the rectangle
Perimeter of Triangle Perimeter = a + b + c a, b and c are the lengths of the three sides of the triangle
Perimeter of Circle Perimeter = πd = 2πr π = 22/7 = 3.14159 (approx)
r = Radius of the circle
d = Diameter of the circle
Perimeter of Parallelogram Perimeter = 2 (l + b) l = Length of the parallelogram
b = Breadth of the parallelogram
Perimeter of Trapezium Perimeter = a + b + c + d a, b, c, d are the lengths of the four sides of the trapezoid
Perimeter of Rhombus Perimeter = 4a a = Length of each side of the rhombus
Perimeter of Kite Perimeter= 2a + 2b a = Length of each side of the first pair
b = Length of each side of the second pair

Additional Area And Parameter Formulas

Some additional formulas to calculate area and perimeters are as under:

Area of a Triangle of Given Sides – a, b, c Area = √ [s (s – a) (s – b) (s – c)] s = Semi-perimeter of the triangle = (a + b + c)/2
Area of an Isosceles Triangle Area = (base x height)/2
base = b
height = √(a2 − b2/4)
a = Length of each of the equal sides of the isosceles triangle
b = Length of the base
Perimeter of an Isosceles Triangle Perimeter = 2a + b a = Length of each of the equal sides of the isosceles triangle
b = Length of the base
Area of an Equilateral Triangle Area = (√3 x a2)/4 a = Length of each side of the equilateral triangle
Perimeter of an Equilateral Triangle Perimeter = 3a a = Length of each side of the equilateral triangle
Perimeter of a Semi-Circle Perimeter = πr + d = 3πr r = Radius of the circle
d = 2r = Diameter of the circle
Area of a Semi-Circle Area = (πr2) x (1/2) r = Radius of the circle

So, now you are aware of the common mensuration formulas that you must have at your fingertips. Memorize them by heart. Make sure you are aware of what the variables in the area formulas and perimeter formulas mean. Solve a sufficient number of practice questions to master the application.

Mensuration based Example:

Q1: The length, breadth and height of a room are in the ratio 3:2:1. If the breadth and height are halved while the length is doubled, then the total area of the four walls of the room will
  1. Remain the same
  2. Decrease by 13.64%
  3. Decrease by 15%
  4. Decrease by 18.75%
  5. Decrease by 30%
Solution: In the present case, let Length = l = 3x, Breadth = b = 2x, Height = h = x
Then, Area of four walls = 2 (l + b) h = 2(3x + 2x) x = 10x2.
Now as Length gets doubled = 6x, Breadth halved = x, Height halved = x/2.
New area of four walls = 2 (6x + x) x/2 = 7x2.
Thus there is a decrease of 30%.
Hence, fifth option is the answer.