MENSURATION
Areas of Plane Figures
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. .
Triangles
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.
Quadrilaterals
(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 = Side2
(b) Area = ½ x
Diagonal2
[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.
