**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 non-parallel 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 + b-2x), 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 semi-circle 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 =

## Leave a Reply