**Area**

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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 ( πr^{2}) is given by

π x 7 x 7 = 3.14 x 7 x 7 = 153.86 square cm or cm^{2}

** 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 cm^{2}

**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 cm^{2}

**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.

**(i)** Area = πr^{2}

**(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.

Area of sector = | θ | X πr^{2} |

360° |

**Area of Circular Ring**

**(i)** Area = π (R^{2}– r^{2})

**(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°.

**(i)** Area = Length × Breadth = L × B

**(ii)** Perimeter = 2(L + B)

(iii) Diagonal (d) = | L^{2} + B^{2} |

**(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°.

(i) Area = (side)^{2} = a^{2} or | 1 | d^{2} |

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.

**(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.

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

2 | 2 |

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.

The opposite angles in a rhombus are equal but they are not right angle.

(i) Area = | 1 | × d_{1} × d_{2 }or Area = base × height |

2 |

**(ii)** Perimeter = 4a

**(iii)** 4a^{2} = d_{1}^{2} + d_{2}^{2}

where, a = side, d_{1} and d_{2} 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) a^{2}

(ii) Each exterior angle = | 360° |

n |

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

(iv) Area of regular Pentagon = | 6a^{2} | √3 |

4 |

(v) Area of regular Hexagon = | 5a^{2} | √3 |

4 |

(vi) No. of diagonals = | { | n(n – 1) | – n | } |

2 |

**Mensuration – Area Formulas and Short Tricks**