Formulas and Short Tricks for Area

Area                                                                                                             « « Go back to Practice exercise


The amount of space inside the boundary of a flat (2-dimensional) object such as a rectangle, triangle, or circle is called the area of that particular figure or we can say area of a particular figure is a measure associated with the part of plane enclosed in the figure. Area is measured in square unit.

For example : If radius of circle is 7 cm then area of circle ( πr2) is given by

π x 7 x 7 = 3.14 x 7 x 7 = 153.86 square cm or cm2

Another example : if length (l) of a rectangle is 8 cm and breadth (b) is 6 cm, then area of rectangle is given as

l × b = 8 × 6 = 48 sq cm or cm2


Perimeter

Perimeter is the length of border around any enclosed plane. Therefore, sum of the sides of a plane figure is the perimeter of that particular figure.

Unit of perimeter is same as the unit of sides of a given figure.

For example : If radius of circle is 7 cm then perimeter of circle ( Circumference (perimeter) = 2πr ) is given by

2πr = 2 x 3.14 x 7 = 43.96 sq cm or cm2

Another example : if the sides of a triangle are 8 cm, 6 cm and 7 cm, respectively, then perimeter of that particular triangle is given as

8 cm + 6 cm + 7 cm = 21 cm


Important Formulae of Circles

  • Area of Circle

It is a plane figure enclosed by a line on which every point is equally distant from a fixed point (centre) inside the curve.

Mensuration - Area Formulas and Short Tricks

(i) Area = πr2

(ii) Circumference (perimeter) = 2πr

(iii) Diameter = 2r

 

  • Sector of a Circle or Area of sector of circle

The area enclosed between the arc , the two radii and the centre of the circle  is called the sector of a circle. Here, the shaded area is the sector of a circle.Mensuration - Area Formulas and Short Tricks

Area of sector        =θX πr2
360°
  • Area of Circular Ring

Mensuration - Area Formulas and Short Tricks

(i) Area = π (R2– r2)

(ii) Difference in circumference of both the rings = (2πR – 2πr)

where, R = radius of bigger ring and r = radius of smaller ring.

Mensuration – Area Formulas and Short Tricks


Important Formulae of Quadrilateral 

A figure enclosed by four sides is called a quadrilateral. A quadrilateral has four angles and sum of these angles is equal to 360°

  • Rectangle

Rectangle is parallelogram with equal opposite sides and each angle is equal to 90°.

Mensuration - Area Formulas and Short Tricks

(i) Area = Length × Breadth = L × B

(ii) Perimeter = 2(L + B)

(iii) Diagonal (d) =L2 + B2

(iv) Area of 4 walls of rectangular room = 2 × (L + B) × h

where, L = Length, B = Breadth, h = Height

Note : The diagonals of a rectangle are of equal lengths and they bisect each other.

  • Square

Square is a parallelogram with alll 4 sides equal and each angle is equal to 90°.

Mensuration - Area Formulas and Short Tricks

(i) Area = (side)2 = a2   or1d2
2

(ii) Perimeter = 4 × side = 4a

where, a = side, d = diagonal

Note : The diagonals of a square are equal and they bisect each other at right angles.

Mensuration – Area Formulas and Short Tricks

  • Parallelogram : It is a quadrilateral with opposite sides parallel and equal.

Opposite angles are equal in a parallelogram but they are not right angle.

Mensuration - Area Formulas and Short Tricks

(i) Area = Base × Height = b × h

(ii) Perimeter = 2 (a + b)

Note : Opposite angles are equal in a parallelogram but they are not right angle. Each diagonal of the parallelogram divides it into two triangles of equal area.

  • Trapezium

Trapezium is a quadrilateral with any one pair of opposite sides parallel.

Mensuration - Area Formulas and Short Tricks

Area  =1 (Sum of the parallel sides) × Height =1(a + b)h
22

where, a and b are parallel sides and h is the height or perpendicular distance between a and b.

  • Rhombus

Rhombus is a parallelogram with all 4 sides equal.

Mensuration - Area Formulas and Short TricksThe opposite angles in a rhombus are equal but they are not right angle.

(i) Area =1 × d1 × d2     or     Area = base × height
2

(ii) Perimeter = 4a

(iii) 4a2 = d12 + d22

where, a = side, d1 and d2 are diagonals.

Note : A rhombus has unequal diagonals and they bisect each other at right angles.

  • Regular Polygon : A polygon is called pentagon, hexagon, octagon, nanogon and decagon according as it contains 5, 6, 7, 8, 9 and 10 sides, respectively.

If each side of a regular polygon of n sides = a, then
(i) Area of regular octagon = 2(2 + 1) a2

(ii) Each exterior angle =360°
n

(iii) Each interior angle = 180° – Exterior angle

(iv) Area of regular Pentagon =6a2√3
4
(v) Area of regular Hexagon =5a2√3
4
(vi) No. of diagonals ={n(n – 1) – n}
2

Mensuration – Area Formulas and Short Tricks