Mensuration is the branch of geometry that deals with the measurement of length, area and volume. We have studied properties of plane figures till now. Here, in addition to areas of plane figures, we will also look at surface areas and volumes of “solids.” Solids are objects, which have three dimensions (plane figures have only two dimensions). .

Let us briefly look at the formulae for areas of various plane figures and surface areas and volumes of various solids. .

The area of a triangle is represented by the symbol D. For a triangle ABC, the three sides are represented by a, b and c and the angles opposite to these sides are represented by A, B and C respectively. .

(i) For any triangle in general,

(a) When the measurements of three sides a, b, c are given,

Area = where

s =

This is called Hero’s formula. .

(b) When base (b) and altitude (height) to that base are given,

Area = ½ x base x altitude = ½ b.h .

(c) Area = ½ ab . sin C = ½ bc.sinA = ½ ca.sinB .

(d) Area = where R is the circumradius of the triangle.

(e) Area = r.s where r is the inradius of the triangle and s, the semi-perimeter. .

Out of these five formulae, the first and the second are the most commonly used and are also more important from the examination point of view. .

(ii) For a right angled triangle,

Area = ½ X Product of the sides containing the right angle

(iii) For an equilateral triangle

Area = where “a” is the side of the triangle

The height of an equilateral triangle =

(iv) For an isosceles triangle

Area = where “a” is length of each of the two equal sides and b is the third side.

(i) For any quadrilateral

Area of the quadrilateral = ½ x One diagonal x Sum of the offsets drawn to that diagonal. .

(ii) For a cyclic quadrilateral where the four sides measure a, b, c and d respectively,

Area = where s is the semi-perimeter, i.e., s = (a + b + c + d)/2

(iii) For a trapezium

Area of a trapezium = ½ x Sum of parallel sides x Distance between them. .

(iv) For a parallelogram

(a) Area = Base x Height

(b) Area = Product of two sides x Sine of the included angle. .

(v) For a rhombus

Area = ½ x Product of the diagonals

Perimeter = 4 x Side of the rhombus

(vi) For a rectangle

Area = Length x Breadth

Perimeter = 2(1 + b), where 1 and b are the length and the breadth of the rectangle respectively

(vii) For a square

(a) Area = Side^{2}

(b) Area = ½ x
Diagonal^{2}

[where the diagonal = x side]

Perimeter = 4 x Side

(viii) For a polygon

(a) Area of a regular polygon = ½ x Perimeter x Perpendicular distance from the centre of the polygon to any side

Please note that the centre of a regular polygon is equidistant from all its sides)

(b) For a polygon which is not regular, the area has to be found out by dividing the polygon into suitable number of quadrilaterals and triangles and adding up the areas of all such figures present in the polygon.