Arithmetics various set of numbers

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1 2 3 4 5 6 7 8

Bog'liq
Inha-math-info-book

Particular cases

Isosceles triangle:



;

2

c

a

h

c











;

c

c

h

c

a

c

r

h

a

R







2

c

a

c

S





Right triangle:



;

2

b

c

b

a

c

a











;

2

c

ab

h

b

a

h









;

c

b

a

r

c

R









;

b

a

R

r













c

b

a

r

R

;



;

2

rR

r

Sin

c

ctg

a

ch

ab

S









xy



;



a

b

c

d















;



a

b

x

y















;

l

is bisector.

equilateral:



;

3

r

R

a

r

a

R





;

3

R

r

R

r

h







a

S



;

4.

Some other relations in triangles:

pq

q

p

q

c

p

a

x







Trapezium



2

c

a

m





is midline;











Sin

f

l

mh

S

;



;

2

S

S

S

S











;

S

S

S





a

h





x

h





b

a

h

b

x

b

a

h

a

y













АЕ=ЕC, BF=FD

MN

p

n

m

b

a















n

m

q

p

x

b

x





y

x

m

n

p

q

2

m

q

y

p

n

x







2

b

p

m

b

a

n







;

b

a

OB

OD

OC

AO



,

h

MN

h

b

a

S

ABCD













2

b

a

h

c

b

a

x











;

h

b

a

Sin

d

S







b

a

b

d

a

c

ab

l







;



b

a

b

c

a

d

ab

f







;



rectangular

trapezium

2

b

a

f

l







;

Parallelogram



2

2

f

l

b

a







;









Sin

b

a

S

;



b

a

h

b

h

a

Sin

f

l

S















;



tan

2

a

b

S







;



DC

PQ

BC

MN

S

S

;





)

(

2

c

b

b

a





Rhombus



;











2

f

l

a



f

l

h

a

r

a

r

p

S

















;









Sin

a

S

;



r

h



;





aSin

f

aCos

l



;

Circumscribed quadrangle



d

b

c

a







;



r

d

b

r

c

a

pr

S

)

(

)

(









;

d

c

b

a

p





;



d

c

b

a

S





;









d

p

c

p

b

p

a

p

S





;

Inscribed quadrangle













180









;



f

l

bc

ad







;









d

p

c

p

b

p

a

p

S





;









bc

ad

bd

ac

cd

ab

S

R





;

Polygons

The sum of inner angles =









2

n

;

The sum of exterior angles =



;

Number of diagonals =







n

n

;

Regular polygons

inner angle –





n

n





;

exterior angle

–



;

1

a

R

r





;

180

4 tan

n

n a

n a

S

pr

r

n



















;

tan

a

R Sin

r

n

n











.

Particular cases

Pentagon:



The sum of inner angles



540

;



Inner angle:



108

;



Exterior angle:



;











r

R

a

;















r

a

R

;















a

R

r

;



a

d





,

d

is diagonal;



5

2













r

a

R

S

;

Hexagon:



The sum of inner angles



720

;



Inner angle:



120

;



Exterior angle:



;



2

r

R

a



;















a

R

r

;



a

d



,

a

R

d



;



2

r

a

R

S



;

Octagon:



The sum of inner angles



1080

;



Inner angle:



135

;



Exterior angle:



;















r

R

a

;











r

a

R

;















a

R

r

;







;













a

R

d

a

d

a

d

;





















r

a

R

S

;

Angles in a circle

































180









x









y











PM

PN

PK

PL











c

c

b

a





BS

AS



BSO

ASO







Arc, sector, segment, ring

Area :

2

D

R

S









;

Length of arc:

180

arc

R

l

 





Area of sector:

sec

360

t

R

S

 





Area of segment:





360

;

segm

R

S

R Sin

R l

a

ah

 

















Cutting area:









360

2

cut

R

S

R

Sin

Sin



 











Area of ring:





2

ring

AB

S

R

r











 







Straight line

straight line, passing from the points





1

y

x

A

;

and





2

y

x

A

;

x

x

x

x

y

y

y

y







;

1

x

x

y

y

k





b

x

k

y





;

2

b

x

k

y





Equation of straight line, passing from the point





, y

x

A

)

(

x

x

k

y

y







;



tan

1

k

k

k k







 

;



Sign of parallelism:

1

k



;



Sign of perpendicularity:

2

1







k

k

;



Sign of intersection:

k

k



;



b

kx

y





;

tan





;



Distance from the point

)

(

0

y

x

A

straight line







c

by

ax

0

b

a

c

by

ax

d





;



Distance between parallel lines:

2

2

1

b

a

c

c

h







;





b

y

a

x

Circle

Length of circle:

D

R

С









;

Equation of circle with center

 

b

a;

and with radius



 



2

R

b

y

a

x









;

Vectors





0

y

x

A

;

is beginning of the vector;





1

y

x

B

;

is the end of the vector;

Coordinates:





1

y

y

x

x

AB





;

Absolute value:



 



1

y

y

x

x

AB









;

Unit vectors



)

;

(

;

(





j

i

;





j

;





j

i

;

j

y

i

x

y

x

a





)

;

(

;



e

is unit vector,

















2

y

x

y

y

x

x

e

;

Arithmetic operations on vectors



 



1

y

x

b

y

x

a

;







y

y

x

x

b

a







;



y

y

x

x

b

a











;







y

x

a







;



 

a

a

a

a

a

a

a







Scalar product:

а) in case, if vectors are given by coordinates:

1

2

1

y

y

x

x

b

a











;

б) in case, if vectors are given by their absolute values:





b

a

Cos

b

a

b

a









;

3) Condition of parallelism:

1

2

1

y

y

x

x



;

4) Condition of perpendicularity:





b

a

Angle between vectors:





1

y

x

y

x

y

y

x

x

b

a

b

a

b

a

Cos











;

SOLID GEOMETRY

Poluhedrons

Prism



Lateral surface area:

lat

base

S

P

l





;



Total area:

2

t

l

b

S

S

S

 

;



Volume:

. .

b

p c

V

S

h

S

l



 





Number of diagonals:

)

(



n

Here

. .

p c

S

is perpendicular cut.

Particular cases:

а) Cuboid:



Area of lateral surface:





l

S

ac

bc





;



Area of total surface:





t

S

ab

ac bc





;



Volume:

abc

V



;



2

c

b

a

d





;

b) Cube:



Area of lateral surface:





l

S

ac bc





;



Area of total surface:

6

t

S

a



;



Volume:

3

a

V



;



3

a



;

2

a

R

a

r



;

Pyramid



Total area:

t

b

l

S

S

S





3

t

V

S

r



;



b

l

S

S Cos





;



Volume:

3

b

V

S

h





b

V

S r



;

2) Regular pyramid

l is generatrix, f is apothem.



b

P

n a

 

;

b

S

n a r

  

;



2

b

l

b

S

S

P

f

Cos





 

;



t

l

b

S

S

S

 

;



3

b

V

S

h





;



r

a

R

















;

2

h

R

l





;

2

h

r

f





;

3) Trancated pyramid







2

l

S

P

P

f







;



2

t

l

S

S

S

S

 



;







1

S

S

S

S

h

V





;

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