Shortcuts in Quantitative Aptitude for Competitive Exams – Mensuration
Shortcuts in Quantitative AptitudeReasoningEnglish
MENSURATION
Mensuration is the science of measurement of the lenghts of lines, areas of surfaces and volumes of solids.
PERIMETER
Perimeter is sum of all the sides. It is measured in cm, m, etc.
AREA
The area of any figure is the amount of surface enclosed within its boundary lines. This is measured in square unit like cm2, m2, etc.
VOLUME
If an object is solid, then the space occupied by such an object is called its volume. This is measured in cubic unit like cm3, m3, etc.
Basic Conversions :

 1 m=10 dm
 1 dm=10 cm
 1 cm=10mm
 1 m= 100 cm= 1000 mm
 1 km= 1000m

 km = miles
 1 mile = 1.6km
 1 inch = 2.54 cm

 100 kg = 1 quintal
 10 quintal = 1 tonne1
 kg= 2.2 pounds (approx.)

 1 litre = 1000 cc
 1 acre = 100 m^{2}
 1 hectare = 10000 m^{2 }(100 acre)
PART I : PLANE FIGURES
TRIANGLE
Perimeter (P) = a + b + c
Area (A) =
where s = and a, b and c are three sides of the triangle.
Also, A = ;
where
b —» base
h —» altitude
EQUILATERAL TRIANGLE
Perimeter=3a
A =
where
a —» side
RIGHT TRIANGLE
A = and
h^{2} = p^{2} + b^{2} (Pythagoras triplet)
where
p —» perpendicular
b —» base
h —» hypotenuse
RECTANGLE
Perimeter = 2 (l +b)
Area = t × b;
where l —> length
b —> breadth
Shortcut Approach
If the length and breadth of a rectangle are increased by a% and b%, respectively, then are will be increased by
SQUARE
Perimeter = 4 x side = 4a
Area = (side)^{2 }= a^{2 };
where
a —> side
PARALLELOGRAM
Perimeter = 2 (a + b)
Area = b x h;
where
a —> breadth
b —> base (or length)
h —> altitude
RHOMBUS
Perimeter = 4a
Area =
where a —> side and d_{1}, and d_{2 }are diagonals.
IRREGULAR QUADRILATERAL
Perimeter=p + q+r + s
Area =
TRAPEZIUM
Perimeter = a + b + m + n
Area =
where (a) and (b) are two parallel sides;
(m) and (n) are two nonparallel sides;
h —» perpendicular distance between two parallel sides.
AREA OF PATHWAYS RUNNING ACROSS THE MIDDLE OF A RECTANGLE
A = a (l+ b) – a2;
where
l —» length
b —» breadth,
a —» width of the pathway.
Pathways Outside
A=(l + 2a)(b+2a) lb;
where
l —> length
b —> breadth
a —> width of the pathway
Pathways Inside
A = lb – (l – 2a) (b – 2a);
where
l —> length
b —> breadth
a —> width of the pathway
Shortcut Approach
If a pathway of width x is made inside or outside a rectangular plot of length l and breadth b, then are of path way is
2x(l + b + 2x), ifpathismade outside the plot.
2x(l + b2x), ifpathismade inside the plot.
If two paths, each of width x are made parallel to length (l) and breadth (b) of the rectangular plot in the middle of the plot, then
area of the paths is x(l+b x)
CIRCLE
Perimeter (Circumference) = 2πr = πd
Area = πr²;
where
r—>radius
d —> diameter
and π = or 3.14
Shortcut Approach
The length and breadth of a rectangle are l and b, then are of circle of maximum radius inscribed in that rectangle is
SEMICIRCLE
Perimeter = πr+ 2r
Area = × πr²
Shortcut Approach
The are a of the largest triangle incribed in a semicircle of radius r isequal to r².
SECTOR OF A CIRCLE
Area of sector OAB = × πr²
Length of an arc (l) = × 2πr
Area of segment = Area of sector – Area of triangle
OAB =
Perimeter of segment = length of the arc + length of segment
AB =
RING
Area of ring =
PART II SOLID FIGURE
CUBOID
A cuboid is a three dimensional box.
Total surface area of a cuboid = 2 (lb + bh + lh)
Volume of the cuboid = lbh
Area of four walls = 2(1 + b) × h
Shortcut Approach
If length, breadth and height of a cuboid are changed by x%, y% and z% respectively, then its volume is increased by
Note: Increment in the value is taken as positive and decrement in value is taken as negative. Positive result shows total increment and negative result shows total decrement.
CUBE
A cube is a cuboid which has all its edges equal.
Total surface area of a cube = 6a² Volume of the cube = a^{3}
RIGHT PRISM
A prism is a solid which can have any polygon at both its ends.
Lateral or curved surface area = Perimeter of base x height
Total surface area = Lateral surface area + 2 (area of the end)
Volume = Area of base x height
RIGHT CIRCULAR CYLINDER
It is a solid which has both its ends in the form of a circle.
Lateral surface area = 2πrh
otal surface area = 2πr (r + h)
Volume = πr²h;
where r is radius of the base and h is the height
PYRAMID
A pyramid is a solid which can have any polygon at its base and its edges converge to single apex.
Lateral or curved surface area
= (perimeter of base) x slant height
Total surface area = lateral surface area + area of the base
Volume = (area of the base) × height
RIGHT CIRCULAR CONE
It is a solid which has a circle as its base and a slanting lateral surface that converges at the apex.
Lateral surface area = πrl
Total surface area = πr (1 + r)
Volume = πr²h;
Where
r : radius of the base
h : height
l : slant height
SPHERE
It is a solid in the form of a ball with radius r.
Lateral surface area = Total surface area = 4πr²
Volume = ;
where r is radius.
HEMISPHERE
It is a solid half of the sphere.
Lateral surface area = 2πr²
Total surface area = 3πr²
Volume = ;
where r is radius 3
Shortcut Approach
If side of a cube or radius (or diameter) of sphere is increased by x%,then its volume increses by
Shortcut Approach
If in a cylinder or cone, height and radius both change by x%, then its volume changes by
FRUSTUM OF A CONE
When a cone cut the left over part is called Curved surface area =πl(r_{1} + r_{2})
Total surface area
where
Volume =
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