Quantitative Aptitude Mensuration Study Material
In mensuration we proceed to determine the lengths of lines, areas of plane surface* and volumes of solids. First of all we shall discuss the areas of two dimensional figures (2-D) like rectangle, square, triangle, circle, parallelogram, rhombus and trapezium and their perimeters. We shall also study some shoit-cut methods to solve such problems.
Let us begin this topic by having a sincere look at the basic definition.
Rectilinear Figures : A figure made up of line segments is known as a rectilinear figure. The line segments forming the figure are called the sides of the figure.
A rectilinear figure having no free ends is said to be a closed figure.
A rectilinear figure is said to be simple if no two sides of it intersect except at a common appoint. Therefore, a rectangle, a square, a triangle, a quadrilateral, a parallelogram, a rhombus and a trapezium are all simple closed rectilinear figures or two dimensional figures.
Region: The part of the plane enclosed by a simple closed figure is called a region. . – •
Area : The magnitude of a plane region enclosed by a simple closed figure is called its area.
Standard Unit of Area : A square centimetre is the area of the region formed by a square of side 1 cm.
A square centimetre (written as cm2 or sq. cm.) is a standard unit of area.
Other standard units of area and their inter- relationships are listed below for ready reference:
1 square centimetre (cm2)
= 1cm x 1cm = 10mm x 10mm = 100 sq. mm
1 square decimetre (dm2)
= 1 dm x 1 dm = 10 cm x 10cm = 100 sq.cm.
1 square metre (m2)
= lmx lm = 10dmx 10dm = 100sq.(dm2)
=100cm x 100cm = 10,000(cm2)
1 acrc= 100(m2)
1 hectare= 100 acres
1 hectares 10,000(m2)
100 hectares = 1km2
Perimeter: Perimeter of a plane figure is the measure of the length of its boundary.
Unit of Perimeter: The unit of perimeter is same as the unit of length. i.e centimetre (cm), metre (m), kilometre (km) etc.
The inter-relationshjp between different units of measurement of length are listedd below for ready reference:
1 centimetre(cm)=10 millimeters (mm)
1 decimetre (dm)= 10 centimeters
1 metre(m)= 10 decimeters = 100 centimetres =1000 millimeters
1 decametre (dam)=10 meters= 1000 centimetres
1 hectometre (hm) = 10 decametres = 100 metres
1 kilometre (km) = 1000 metres = 100 decametres = 10 hectometres ‘
1 myriametre= 10 kilometres.
In mensuration it is very important to know and understand the basic formulae.
So, we shall I first see the basic formulae:
In a rectangle all the angles are of 90° and opposite sides equal. The figure ABCD is a rectangle
For a rectangle of length = l
and breadth = b ,we have
(i) Perimeter = 2 (Length + Breadth) = 2 (1 + b) units.
(ii) Diagonal = √l2 + b2
In the figure, AC and BD are diagonals.
(iii)Area = 1 x b. sq. units
(iv) Length = Area/Breadth
and Breadth = Area/ Length
(v) If l, b and h denote the length, breadth and height of a room, then Area of 4 walls = 2 (1 + b) x h. sq.
(vi) Diagonal of the room = √l2+b2 +h2 units
Square: In a square, all sides are equal and all angles are of 90°. The figure given below is a square.
For a square, each of whose sides measures a units, we have
- Area = a2 square units.
- Side of the square = √area
- Diagonal of the square = √2
Area of the square = [1/2( diagonal)2 ]
- Perimeter of the square = 4a units.
Perimeter and Area of a Triangle
- When the base and the corresponding altitude or height of a triangle are given then
(iii) Right-Angled Triangle : A triangle with one angle equal to 90°.
Let b be the base, p be the perpendicular and h be the hypotenuse of a right angled triangle. Then
- Perimeter = b + p + h.
- Area 1/2xBasex Height = 1/2bp
- Hypotenuse = √b2 + p2 [By Pythagoras theorem, h2 = b2 + p2]
(iv) Isosceles Triangle : A triangle with any two sides equal. In an isosceles triangle, opposite angles are equal.
(vi) Equilateral Triangle: A triangle with all sides equal. For an equilateral triangle, each of whose sides is a, we have
(a) Perimeter = 3a (b) Height = √3/a (c) Area= √3/4 a2
Areas and Perimeters of a Quadrilateral, a Parallelogram, a Rhombus and a Trapezium.
If P1 and P2 be the perpendicular distances of the diagonal AC of a quadrilateral ABCD from the vertices B and D respectively, then
Parallelogram : A quadrilateral with opposite sides parallel and equal.
For a Parallelogram with adjacent sides a and b we have
- Perimeter = 2 (a + b) = 2 (Sum of adjacent sides).
Area = Base x Height
Rhombus: A rhombus is a parallelogram having all sides equal. The diagonals of a rhombus bisect each other at right angles.
If d1 and d2 are the diagonals of a rhombus, we have:
Trapezium : A trapezium is a quadrilateral two of whose sides are parallel. A trapezium whose non-parallel sides are equal is known as an isosceles trapezium.
For a trapezium, whose parallel sides are a and b and h is the perpendicular distance between them we have :
Area of a Circle, Sector and Segment of a Circle
Circle: A circle is the locus of a point which moves in a plane in such a way that it is equidistant from a fixed point.
The fixed point is called the centre and the constant distance is known as the radius of the circle.
The perimeter of a circle is generally known as its circumference.
Chord of a Circle: A line segment AB whose end points lie on the circle is called a chord of the circle.
A chord passing through the centre of a circle is the longest chord, called the diameter. Clearly, diameter = 2 x radius.
Arc of a Circle: A chord of a circle, other than the diameter, divides the circle into two parts, one smaller than the other. Each part is called an arc.
The smaller part is known as the minor arc and the greater part is known as the major arc.
In the above figure, A X B is the minor arc and AYB is the major arc.
Semi Circle : A diameter of a circle divides it into two equal parts. Each part is called a semicircle.
A diameter of a circle divides the circular region into two equal parts. Each part is called a semicircular region.
Sector of a Circle : A sector of a circle is a region enclosed by an arc of the circle and the two radii to its end points.
Thus in the given figure, region OAB is a sector.
The angle subtended by the arc of a sector at the centre is the sector angle or the central angle of the arc.
The degree measure of the central angle of an arc is called the degree measure of the arc.
Minor Sector: A sector of a circle is called a minor sector if the minor arc of the circle is a part of its boundary.
Major Sector: A sector of a circle is called a major sector it the major arc of the circle is a part of its boundary. Remarks:
— A minor sector has an angle 0°, say, subtend at the centre of the circle whereas a major sector has no angle.
— The sum of the arcs of major and minor sectors of a circle is equal to the circumference of the circle.
— The sum of the areas of major and minor sectors of a circle is equal to the area of the circle.
— The boundary of a sector consists of arc of the circle and the two radii.
Segment of a Circle: A segment of a circle is the region enclosed by an arc of the circle and the chord.
Concentric Circles : Circles having the same centre but with different radii are said to be concentric circles.
Path Around a Garden, Verandah Around Aroometc
Some Important Points
Case-I : When the path or verandah is outside the garden or room, surrounding it:
Area of Path = 2(Width of Path) x [Length + Breadth of garden + 2 x (Width of Path)]
Case-II: When the path is inside the garden, surrounded by it:
Area of Path = 2 (Width of Path) x [Length + Breadth of garden – 2 (Width of Path)]
Case-Ill: When paths are crossing each other.
Area of the Path = [(Width of Path) (Length + Breadth of garden – Width of Path)]