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1. Important Formulas

2. Solved Examples

Important Formulas - Area

1.

Pythagorean Theorem (Pythagoras' theorem)

In a right angled triangle, the square of the hypotenuse is equal to thesum of the squares of the other two sides

c2 = a2 + b2 where c is the length of the hypotenuse and a and b are thelengths of the other two sides

2. Pi is a mathematical constant which is the ratio of a circle's circumference to itsdiameter. It is denoted by

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3. Geometric Shapes and solids and Important Formulas

Geometric Shapes Description Formulas

Rectangle

l = Length

b = Breadth

d= Length ofdiagonal

Area = lb

Perimeter = 2(l + b)

d =

Square

a = Length of aside

d= Length ofdiagonal

Perimeter = 4a

d =

Parallelogram

b and c aresides

b = base

h = height

Area = bh

Perimeter = 2(b + c)

Rhombus

a = length ofeach side Area = bh(Formula 1 for area)

3.14 22

7

+l2 b2

Area = =a21

2d2

a2

1


b = base

h = height

d1, d2 are thediagonal

Area = (Formula 2 for area)

Perimeter = 4a

Triangle

a , b and c aresides

b = base

h = height

(Formula 1 for area)

(Formula 2 for area -

Heron's formula)

Perimeter = a + b + c

Radius of incircle of a triangle of area A =

EquilateralTriangle

a = side

Perimeter = 3a

Radius of incircle of an equilateral triangle of side a =

Radius of circumcircle of an equilateral triangle of side a =

1

2d1 d2

Area = bh1

2

Area = S(S a)(S b)(S c)

where S is the semiperimeter = a+ b + c

2

where S is the semiperimeter = A

S

a+ b + c

2

Area = 3

4a2

a

2 3

a

3


Base a is parallele to base b

Trapezium(Trapezoid inAmericanEnglish)

h = height

Circle

r = radius

d = diameter

d = 2r

Sector ofCircle

Area = (a+ b)h1

2

Area = = r21

4d2

Circumference = 2r = d

= Circumference

d

Area, A =

(if angle measure is in degrees)

360r2

(if angle measure is in radians)1

2r2

Arc Length, s =

r (if angle measure is in degrees)

180

r (if angle measure is in radians)


r = radius

= centralangle

Ellipse

Major axislength = 2a

Minor axislength = 2b

Area = ab

RectangularSolid

l = length

w = width

h = height

Total Surface Area = 2lw + 2wh + 2hl = 2(lw + wh + hl)

Volume = lwh

Plese note that in the radian system for angular measurement,

2 radians = 360

1 radian = 180

1 = radians

180

Hence,

Angle in Degrees = Angle in Radians 180

Angle in Radians = Angle in Degrees

180

Perimeter 2+a2 b2

2


Cube

s = edge

Total Surface Area = 6s2

Volume = s3

RightCircularCylinder

h = height

r = radius ofbase

Lateral Surface Area = (2 r)h

Total Surface Area = (2 r)h + 2 ( r2)

Voulme = ( r2)h

Pyramid

h = height

B = area of thebase

Total Surface Area = B + Sum of the areas of the trianguar sides

Volume = Bh1

3


RightCircular Cone

h = height

r = radius ofbase

Sphere

r = radius

d = diameter

d = 2r

4. Important properties of Geometric Shapes

I. Properties of Triangle

i. Sum of the angles of a triangle = 180

ii. Sum of any two sides of a triangle is greater than the third side.

iii. The line joining the midpoint of a side of a triangle to thepositive vertex is called the median

iv. The median of a triangle divides the triangle into two triangleswith equal areas

v. Centroid is the point where the three medians of a trianglemeet.

vi. Centroid divides each median into segments with a 2:1 ratio

vii. Area of a triangle formed by joining the midpoints of the sides ofa given triangle is one-fourth of the area of the given triangle.

Lateral Surface Area =r = rs +r2 h2

where s is the slant height = +r2 h2

Total Surface Area = r + = rs ++r2 h2 r2 r2

Surface Area = 4 = r2 d2

Volume = = 4

3r3

1

6d3


viii. An equilateral triangle is a triangle in which all three sides areequal

ix. In an equilateral triangle, all three internal angles are congruentto each other

x. In an equilateral triangle, all three internal angles are each 60

xi. An isosceles triangle is a triangle with (at least) two equal sides

xii. In isosceles triangle, altitude from vertex bisects the base.

II. Properties of QuadrilateralsA. Rectangle

i. The diagonals of a rectangle are equal and bisect each other

ii. opposite sides of a rectangle are parallel

iii. opposite sides of a rectangle are congruent

iv. opposite angles of a rectangle are congruent

v. All four angles of a rectangle are right angles

vi. The diagonals of a rectangle are congruent

B. Square

i. All four sides of a square are congruent

ii. Opposite sides of a square are parallel

iii. The diagonals of a square are equal

iv. The diagonals of a square bisect each other at right angles

v. All angles of a square are 90 degrees.

C. Parallelogram


i. The opposite sides of a parallelogram are equal in length.

ii. The opposite angles of a parallelogram are congruent (equalmeasure).

iii. The diagonals of a parallelogram bisect each other.

iv. Each diagonal of a parallelogram divides it into two triangles ofthe same area

D. Rhombus

i. All the sides of a rhombus are congruent

ii. Opposite sides of a rhombus are parallel.

iii. The diagonals of a rhombus are unequal and bisect each otherat right angles

iv. Opposite internal angles of a rhombus are congruent (equal insize)

v. Any two consecutive internal angles of a rhombus aresupplementary; i.e. the sum of their angles = 180 (equal insize)

Other properties of quadrilaterals

i. The sum of the interior angles of a quadrilateral is 360 degrees

ii. A square and a rhombus on the same base will have equalareas.

iii. A parallelogram and a rectangle on the same base and betweenthe same parallels are equal in area.

iv. Of all the parallelogram of given sides, the parallelogram which isa rectangle has the greatest area.

v. Each diagonal of a parallelogram divides it into two triangles ofthe same area


III. Sum of Interior Angles of a polygon

i. The sum of the interior angles of a polygon = 180(n - 2)degrees where n = number of sides Example 1 : Number of sidesof a triangle = 3. Hence, sum of the interior angles of a triangle= 180(3 - 2) = 180 1 = 180 Example 2 : Number of sides ofa quadrilateral = 4. Hence, sum of the interior angles of anyquadrilateral = 180(4 - 2) = 180 2 = 360

Comments(17) Newest

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sri 05 Jun 2013 7:48 AM

need more formulas with clear concepts

Somnath Bera 28 May 2013 1:36 AM

really helpful notes and problem as well...I requested to you to engrave your problemexchequer that could help others those who have sited for public service exams...

vasavi 11 May 2013 10:23 AM

thank u so much.........

ujwala 03 May 2013 7:15 AM

very very nice

shivarudra Mcom hubli 03 May 2013 2:05 AM

thank u very much

shivarudra Mcom hubli 03 May 2013 2:04 AM


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i have learnt many unknown things and thank u very much

GANESH 19 Apr 2013 6:28 AM

GOOD .............

ganesh 19 Apr 2013 6:27 AM

very nice

mahesh kumar 17 Apr 2013 5:50 AM

very good work.......,

Arun kumar R 12 Apr 2013 6:29 AM

I need more formulas related to ssc examination...need more questions and answers formy future reference ....by yours faithfully

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