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Basic Formulas

INDICES

1 a. a. am = am

2 a-m = 1 am

3 am

x an

= am+n

4 am an = am n ( m > n ) or 1 am n ( m < n )

5 (am)n = amn

6 (ab)m = am x bm

7 (ab)m = am bm

8 a0 =1

9 a = a (1/2)

10 3 a = a(1/ 3)

11 na = a(1/ n)

12 nab = na x nb = a(1/ n) x b(1/ n)

13 na b = na nb = a(1/ n) b(1/ n)

14 nam = a(m / n) = (a(1/ n))m = (am)(1/n)

ALGEBRA

1 ( a + b )2

= a2

+ 2ab + b2

2 ( a - b )2 = a2 - 2ab + b2

3 ( x + a ) ( x + b ) = x2 + ( a + b ) x +ab

4 ( x + a ) ( x - a ) = x2 - a2

5 ( a + b )2 + ( a - b )2 = 2 (a2 + b2)

6 ( a + b )2 - ( a - b )2 = 4ab

7 a2 + b2 = (a + b)2 - 2ab

8 a

2

+ b

2

= (a - b)

2

+ 2ab9 (a + b )3 = a3 + b3 + 3ab ( a + b )

10 (a - b )3 = a3 - b3 - 3ab ( a - b )

11 a3 + b3 = ( a + b ) (a2 - ab + b2)

12 a3 - b3 = ( a - b ) (a2 + ab + b2)

13 a3 + b3 + c3 3abc = ( a + b + c ) ( a2 + b2 + c2 ab bc ca)

14 (a + b + c )2 = a2 + b2 + c2 + 2(ab + bc + ca )

15 a

2

+ b

2

+ c

2

= ( a + b + c )

2

- 2(ab + bc + ca )16 if a + b+ c = 0, then a3 + b3 + c3 = 3abc

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Basic Formulas

Basic symbols:

i) A + B = ii) A + = X iii). A B = ? iv). A B

=

Logarithms

1 logamn = logam + logan

2 logam/n = logam - logan

3 loga(m)n = nlogam

4 log101 = 0

5 log1010 = 1

6 logaa = 1

GEOMETRY

1 Area of triangle = x base height (or) (1/2 bh) Unit2

2 Perimeter of a square = 4 x side (or) (4a)

3 Area of square = (side) 2 (or) a2 (or) d2/2 Unit2

4. Perimeter of rectangle = 2 (length + breadth)(or) [2 (l+b)] Unit2

5 Area of rectangle = length breadth (or) (l b) Unit2

6 Area of parallelogram = base height (or) (bh)

7 Area of trapezium = x (sum of parallel sides) (distance between

them)

8 Area of quadrilateral = x (diagonal) (sum of internal perpendiculars on it)

IF RADIUS OF CIRCLE IS R THEN

9 Perimeter of the circle = 2r (or) D

10 Area of the circle = r2 (or) d2/4 Unit2

11 Area of the Hollow Circle = (R+r) (R-r) (or) d t Unit2

IF RADIUS OF SPHERE IS R THEN

12 Lateral Surface area = 4r2

13 Volume = 4/3 r3 Unit3

IF RADIUS OF[1/2 (or) HEME] SPHERE IS R THEN

14 Surface area = 2r2

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Basic Formulas

15 Total Surface area = 3r2

16 Volume = 2/3 r3 Unit3

IF VOLUME OF HOLLOW SPHERE

17 Volume of hollow sphere = 4/3 (R3 r3) Unit 3

IF RADIUS AND HEIGHT OF A CYLINDER ARE R AND H THEN

18 Lateral Surface area = 2rh Unit2

19 Total surface area = 2r (h + r) Unit2

20 Volume = r2h Unit3

21 Volume of Hollow Cylinder = (R2 - r2 ) h (or) D T H Unit3

IF RADIUS IS R, HEIGHT IS H AND SLANT HEIGHT IS L OF A CONE THEN

21 Curved Surface area = rl

22 Total surface area = r (r + l)

23 Volume = 1/3r2h Unit3

IF SIDE IS A, OF A CUBE OF SIDE, THEN

24 Total Surface area = 6a2

25 Lateral Surface area = 4a2

26 Volume = a3 Unit3

IF L, B, H ARE THE SIDES OF A RECTANGULAR PARALLELOPIPED THEN

24 Surface area = 2 (lb + bh + lh)25 Volume = l b h26 Diagonal of a cube of side a = 3a27 Area of regular hexagon A = (33 2 ) (side)228 Area of regular octagon A = 4.84 (side)229 Height of a equilateral triangle = (3 2) (side)30 Area of a equilateral triangle = (3 4) (side)231 Area of isosceles triangles = base 4 4 (side)2 - (base)2

HEXAGONAL

32 Area of the Hexagonal = 6 3 4 a2

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Basic Formulas

TRIGONOMETRY

1 sin2 + cos2 = 1

2 sin2 = 1 - Cos2

4 Sec2 tan2 = 1

5 Sec2 = 1+ tan2x

6 tan2

= Sec2

-17 Cosec2 -Cot2 = 1

8 Cosec2 = 1 + cot2

9 Cot2 = Cosec2 - 1

10 tan = SinCos

11 Cot = CosSin

12 sin( A + B) = SinA. CosB + CosA. sinB

13 sin( A - B) = SinA. CosB - CosA. sinB

14 cos( A + B) = cosA. CosB - sinA. SinB

15 Cos( A - B) = cosA. CosB + sinA. SinB

16 Sin( A + B) + sin( A - B) = 2sinA cosB

17 Sin( A + B) - sin( A - B) = 2cosA sinB

18 Cos( A + B) + cos( A - B) = 2cosA cosB

19 Cos( A + B) - cos ( A - B) = -2sinA sinB

20 Sin( A + B) Sin ( A B) = Sin2A Sin2B

21 Cos (A+B) Cos(A-B) = Cos2A Sin2B

22 Sin2A = 2SinA CosA

23 Tan(A+B) = tanA + tanB (1- tanA . tanB)

24 Tan(A-B) = tanA - tanB (1+ tanA . tanB)

25 Tan 2A = 2tanA 1-tan2A

26 Tan2A = 1-Cos 2A 1+Cos 2A

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Basic Formulas27 Cos2A = Cos2ASin2A (or) 2Cos2A-1 (or) 1- 2Sin2A

28 Cos 2A = 1-2Sin2A

29 Cos 2A = 2Cos2A-1

30 Sin 2A = 2 tanA 1+ tan2A

31 Cos 2A = 1-tan2A 1+ tan2A

32 SinC + SinD = 2Sin [(C+D)2] Cos[(C-D)2]

33 SinC - SinD = 2Cos [(C+D)2] Sin[(C-D)2]

34 CosC + CosD = 2Cos [(C+D)2] Cos[(C-D)2]

35 CosD - CosC = 2Sin [(C+D)2] Sin[(C-D)2]

36 Sin x . Sin y = [ Cos (x-y) Cos (x+y) ]

37 Sin x . Cos y = [ Sin (x+y) + Sin (x-y) ]38 Cos x . Cos y = [Cos (x-y) + Cos(x+y)]

39 A + B = C & A - B = D (or) 2A = C +D(or) A = C+D/2(or) A= C-D/2

SINE FORMULA

40 a /sinA = b / sinB = c/sinC = 2R

NAPION FORUMLA

41 Tan (A-B 2) = [ (a-b) (a+b) ] Cot C/2

42 Tan (B-C 2) = [ (b-c) (b+c) ] Cot A/2

43 Tan (C-A 2) = [ (c-a) (c+a) ] Cot B/2

COSEC FORMUL

44 a2 = b2 + c2 2bc cosA

45 b2 = c2 + a2 2ca cosB

46 C2 = a2 + b2 2ab cosC

47 Sin A/2 = [ ( S-b) (S-c) bc ]

48 Sin B/2 = [ ( S-c) (S-a) ca ]

49 Sin C /2 = [ (S-a) (S-b) ab ]

50 Cos A/2 = [ S (S-a) bc]

CALCULUS

LIMITS

IMPORTANT LIMIT:

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Basic FormulasLt xn - an

1 x 0 x a = n a n-1

Lt sin2 x 0 = 1

Lt3 x a f (x) = f (a)

4 Lt (U V) = Lt U Lt VXa xa xa

Lt (U. V) = Lt U Lt V5 xa xa xa

Lt U/V = Xa

6 Xa Lt VXa

DIFFERENTIATION Lt

7 h 0 f (a + h) = f (a)

` Lt8 x a f (x) = f (a)

Y/n Dy / dxxn n xn-1 diff(x)

Constant 0

1

ex ex diff(x)

ax ax (loga)

x 1

Log ex 1

Log x 1x diff(x)Log a 0

Log ax Log ae xLog e 0

x 1 2x diff(x)

Log x 1 2x diff(x)Sin x Cos x diff(x)Cos x -Sin x diff(x)

Tan x Sec2

x diff(x)cot x -Cosec 2 x diff(x)

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Basic FormulasSec x Sec x tan x diff(x)

cosec x -Cosec x cot x

Sin-1 x 1 / 1- x 2 diff(x)

Cos-1 x - 1 / 1- x 2 diff(x)Tan-1 x 1 / 1+ x 2 diff(x)

Cot-1 x - 1 / 1+ x 2 diff(x)Sec-1 x 1 / xx 2 1 diff(x)

Cosec-1 x - 1 / xx 2 1 diff(x)

dy/dx [K. f (x)] K .f' (X)

Constant function K, C, etc.

dy/dx [ f (x) g (x)] F (x) g' (x) + g (x) f' (x)

dy/dx f ( g (x)) f ' (g (x) ) .g ' (x)Log eA = 0, (or) [(A= e0 )=1] Any Base & Power 0, the

value =1 (or) Constant =1

DIFFERENTIATION FORMULAS

Log Using Method

1. Y = U V = log y = log u + log v

2. Y = U V = logy = log u log v

VU' - UV' (or) (V du/dx - U dv/dx) V2 V2

3. Y = U V W

log y = log u + log v + log w(or)

dy/dx = uvw'+ vwu'+ uwv '

4. uv = vuv -1 u ' + uv (logu)v '

d[c f (x)]

5 dx = c f' (x)

6 y = f [g(x) ]7 y = f (u)

INTEGRATION OF FORMULAS

1 xn .dx = X n+1 (or) 1xn dx (or) dx.x-n = x - n+1

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Basic Formulasn+1 n+1

2 ex .dx = ex + C diff(x)

3 ax dx = ax log a + C

4 dx = x + C

5 d = + C

6 dy = y + C

7 Kdx = kx + C

8 3 dx = 3x +C

9 1x .dx = log (x) + C

[d/dx (log x) = 1/x ] Differention Forms

[d/dx -(log x) = 1/x ] Differention Forms10 sin x .dx = -cos x + C

11 cos x .dx = sin x + C

12 Tan x .dx = log (Sec x) + C

13 Cot x .dx = log sin x + C

14 Sec x .dx = log (Sec x +tan x) + C

15 Cosec x dx = log (cosec x cot x)+ C

16 sec2 x .dx = tan x + C

17 cosec2 x .dx = -cot x + C

18 secx tan x. dx = sec x +C

19 cosecx cotx .dx = -cosec x +C

20 1 1 x2 .dx = sin-1 x + C

21 1 1 + x2 = tan-1 x + C

22 1 x x2 1 .dx = sec-1 x + C

23 f '(x)f(x) .dx = log f (x) + C

24 (ax + b)n.dx = 1a [ (ax +b)n+1n+1] +C

25 (ax + b)-1.dx = 1a log (ax +b) +C

26 dx a +bx = 1b log (a+bx) + C

27 dx 4+gx = 1 g log (4+gx) + C

28 dx a-bx = -1 b log (a-bx) + C

29 dx px +q = 1 p log (px+q) + C

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Basic Formulas30 dx7x-5 = 1 7 log 97x-5) + C

31 dx 3-2x = -12 log (3-2x) + C

32 Cosec (ax+b)cot(ax+b) dx = -1a cosec(ax + b) + C

33 Sec2 (ax+b) dx = 1a tan (ax+b) + C

34 1 (ax + b) dx = 1a log (ax+b) + C

35 eax+b dx = 1a eax+b + C

36 Sin(ax+b) dx = -1a Cos (ax+b) + C

37 Cos (ax+b) dx = 1a Sin (ax+b) + C

38 Cosec2 (ax+b) dx = -1a Cot (ax+b) + C

39 1[1+(ax)2] dx = 1a tan-1(ax) + C

40 1[1-(ax)2] dx = 1a Sin-1(ax) + C

BASIC RULES

41 C f (x) dx = C f (x) dx

42 [ f (x) g (x) ] dx = f (x) dx g (x) dx

SCIENCY FORMULAS

1 Mass = Density Volume.

Density () = Mass Volume. (Kg/m3)

2 Force = Mass x gravity (1 N (or) 1 kg-m/Sec2 , UnitNewton )

3 Weight = Mass Acceleration due to Gravity.

4 Volume = 1m3 (or)1000 Litter.

5 Density () = Mass Volume.

6 Specific Weight (W) = Weight Volume . Unit N/m3 (or)KN/m3.

7 Specific Volume (V) = Volume Mass . M3/kg .

8 Pressure = Force Area . N/m2 (or) KN/m2 (or) MN/n2 .

9 Work = Force Displacement. N meter.

10 Speed (V) = Distance moved Time Taken . Km/hor.

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Basic Formulas11 Power (p) = Work time . N m /Sec. (or) Watt

12 Stress = Load Area (or) (P/L) N/mm2

13 Strain = Change in dimension Original dimension(or)

(l l) (No unite)

14 Youngs modules (E) = Stress Strain (N/mm2)

16 Power (P) = 2NT60 Kw (or) W

17 Acceleration (A) = Rate of Change of Velocity. M/Sec2

18 Velocity (V) = Displacement Time taken. m/Sec

ANGLE IN TO RADION

180 = Radion

1. 180 = 2 2 = 230 = ? Ans :- 30 x x = x

180 62. 180 =

45 = ? Ans :- 45 2 = 1180 4 2 2 2

3. 180 = 60 = ? Ans :- 60 2 = 2

180 3 2 2 24. 180 =

90 = ? Ans :- 90 180 2

5. 180 =

120 = ? Ans :- 120 7180 66. 180 =

135 = ? Ans :- 135 3180 4

7. 180 = 150 = ? Ans :- 150 5

180 68. 180 =

180 = ? Ans :- 180 180 1

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Basic Formulas9. 180 =

210 = ? Ans :- 210 2180 3

10. 180 =

225 = ? Ans :- 225 5180 4

11. 180 = 240 = ? Ans :- 240 4

180 3

12. 180 = 270 = ? Ans :- 270 3180 2

13. 180 = 300 = ? Ans :- 300 5

180 314. 180 =

315 = ? Ans :- 315 7180 4

15. 180 = 330 = ? Ans :- 330 11

180 616. 180 =

360 = ? Ans :- 360 2180 1

RADION IN TO ANGLE

= 180 Angle

1. = 18052 = ? Ans :- 5 180

2 = 450

2. = 18032 = ? Ans :- 3 180

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Basic Formulas2 = 270

3. = 18074 = ? Ans :- 7 180

4 = 315

Special PolygonsSpecial Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid.

Special Triangles - right, equilateral, isosceles, scalene, acute, obtuse.

Polygon Names

Generally accepted names

Sides Name

n N-gon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

10 Decagon

12 Dodecagon

Names for other polygons have been proposed.

Sides Name

9 Nonagon, Enneagon

11 Undecagon, Hendecagon

13 Tridecagon, Triskaidecagon

14 Tetradecagon, Tetrakaidecagon

15 Pentadecagon, Pentakaidecagon

16 Hexadecagon, Hexakaidecagon

17 Heptadecagon, Heptakaidecagon

18 Octadecagon, Octakaidecagon

19 Enneadecagon, Enneakaidecagon

20 Icosagon

30 Triacontagon

40 Tetracontagon50 Pentacontagon

60 Hexacontagon

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Basic Formulas70 Heptacontagon

80 Octacontagon

90 Enneacontagon

100 Hectogon, Hecatontagon

1,000 Chiliagon

10,000 Myriagon

To construct a name, combine the prefix+suffix

Sides Prefix

20 Icosikai...

30 Triacontakai...

40 Tetracontakai...

50 Pentacontakai...

60 Hexacontakai...

70 Heptacontakai...

80 Octacontakai...

90 Enneacontakai...

+

Sides Suffix

+1 ...henagon

+2 ...digon

+3 ...trigon

+4 ...tetragon

+5 ...pentagon

+6 ...hexagon+7 ...heptagon

+8 ...octagon

+9 ...enneagon

Examples:46 sided polygon - Tetracontakaihexagon

28 sided polygon - Icosikaioctagon

However, many people use the form n-gon, as in 46-gon, or 28-gon instead of these names.

pi= = 3.141592...)

90

II Qdt (90+) (90-) I Qdt

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Basic Formulas

(90 To 180) (0 To 90){(Sin & Cosec(+Ve))} {All+ Ve}(180-) (0+) OPP HYP

180 0,360

III Qdt (180+) IV Qdt (360-) ADD (180 To 270) (270To 360)

{(Tan & Cot (+ Ve))} {(Cos & Sec (+Ve))}

(270-) (270+)1. All Silver Tea Cups

270 (or)2. Annai Sathiya Transport Corporation

1. For an angle either (90 , I Qdt) & (270, III Qdt) The Following Change Aer to be Taken Place.RULES:- I ( 90) & III (270) Example:- (180+30)IIIQdt

Sin Cos 1.( +)vd;why; mLj;j Qdtia ghHj;JCos Sin (+ ,-) Nghl Ntz;Lk;.

Tan Cot Example:- (180-30) II Qdt

Cosec Sec 2. (- ) vd;why; mNj Qdt ia ghHj;J

Sec Cosec (+ , -) Nghl Ntz;Lk;. Cot Tan

RULES:-

2. I .In first Qdt All ( Sin , Cos , Tan , Cosec, Sec, Cot) are (+Ve).

ii. In Second Qdt Only (Sin & Cosec) are (+Ve) & Remaining Things are (- Ve).

iii. In Third Qdt Only (Tan & Cot) are (+Ve) & Remaining Things are (-Ve).

iv. In Forth Qdt only (Cos & Sec) are (+Ve) & Remaining Things are (-Ve).

3. Example :

i. Sin (90+) = +Cos iii. Tan 210 (180+30) = +Tan 30ii.Cot (360-) = -Cot iv. Cot 420 (360+60) = +Cot 60

v.Cosec 300 (270+30) = -Sec 30vi.Sec 210 (270-60) = - Cosec 60 (or)

Sec 210 (180+30) = - Sec 30

1. The Circle Is Divided in to four Quitrents namely I , II , III , IV, respectively.

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Basic Formulas2. In the I qdt they angle liese in between 0 to 90 , in the II end Qdt 91 to 180,

In the III ed Qdt 181 to 270 , and ends with IV th Qdt 271to 360.

3. For an angle either { 90 I st Qdt} & {270III Ed Qdt} the following changes are to betaken place.

( 90 I St Qdt ) & ( 270 III Th Qdt)Example:- (180+30) III Qdt

Sin Cos 1.( +)vd;why; mLj;j Qdt ia ghHj;JCos Sin (+ ,-) Nghl Ntz;Lk;.Tan Cot

Cosec Sec 2. (- ) vd;why; mNj Qdt ia ghHj;JSec Cosec (+ , -) Nghl Ntz;LkCot Sec Example:- (180-30) II Qdt

4. For getting sign of the trigonometrically functions the following rulesare followed.

Sin (90-)

Here the angle in 90 & also (-) is occurred .

So it lies in the I Qdt Apply I Qdt Rule.

If it is Sin (90+) Apply I Qdt Rule .

5. Example :

i. Sin (90+) = +Cos iii.Tan 210 (180+30) = +Tan 30ii.Cot (360-) = -Cot iv.Cot 420 (360+60) = +Cot 60

v.Cosec300 (270+30) = -Sec 30vi.Sec 210 (270-60) = -Cosec 60

(or)

Sec 210 (180+30) = - Sec 30

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Basic FormulasGENERAL MATH FORMULAS

AREA FORMULAS

CIRCLE:

r2 (=constant=3.142(approx), and r = radius) RECTANGLE:

ab (a = width and b = height of rectangle)

SQUARE:

a2 (a = width = height of square, all sides equal)

TRIANGLE:

1/2(bh) (b = base width and h = height of triangle)

SURFACE AREA CYLINDER:

2( r2 + rh) (=constant=3.142(approx), r = radius, h =height)

CUBE:

6a2 (a=length of each side of the cube)

RECTANGULAR PRISM:

2(ab + ac + bc) (a, b, and c are the lengths of the 3 sides)

SPHERE:

4r2 (=constant=3.142(approx), and r = radius)VOLUME FORMULAS

CUBE:a3 (a = length = width = height of square, all sides equal)

CONE:

1/3r2h (=constant=3.142(approx), r = radius, h = height) CYLINDER:

r2h (=constant=3.142(approx), r = radius, h = height) PYRAMID:

1/3(bh) (b = base width and h = height of pyramid)

RECTANGULAR PRISM:

abc (a = length, b = width and c = height of rectangular

prism )

PERIMETER: CIRCLE:

d or 2r (=constant=3.142(approx), d = diameter and r = radius)

RECTANGLE:

2(a + b) or a + b + a + b (a = length, b = height of rectangle)

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Basic FormulasWhat is a Polygon?A closed plane figure made up of several line segments that are joined together. The sides do not crosseach other. Exactly two sides meet at every vertex.

Types of PolygonsRegular - all angles are equal and all sides are the same length. Regular polygons are both equiangularand equilateral.Equiangular - all angles are equal.Equilateral - all sides are the same length.

Convex - a straight line drawn through a convex polygon crosses at most two sides.Every interior angle is less than 180.

Concave - you can draw at least one straight line through a concave polygon that crossesmore than two sides. At least one interior angle is more than 180.

Polygon Formulas(N = # of sides and S = length from center to a corner)

Area of a regular polygon = (1/2) N sin(360/N) S2

Sum of the interior angles of a polygon = (N - 2) x 180

The number of diagonals in a polygon = 1/2 N(N-3)The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)

Polygon Parts

Side - one of the line segments that make up the polygon.

Vertex - point where two sides meet. Two or more of thesepoints are called vertices.

Diagonal - a line connecting two vertices that isn't a side.

Interior Angle - Angle formed by two adjacent sides insidethe polygon.

Exterior Angle - Angle formed by two adjacent sidesoutside the polygon.

Special Polygons

Special Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid.


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Basic Formulas

Polygon Names

Generally accepted names

Sides Name

N N-gon3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

10 Decagon

12 Dodecagon


Sides Name

9 Nonagon, Enneagon







18 Octadecagon, Octakaidecagon19 Enneadecagon, Enneakaidecagon

20 Icosagon

30 Triacontagon

40 Tetracontagon

50 Pentacontagon

60 Hexacontagon

70 Heptacontagon

80 Octacontagon

90 Enneacontagon


1,000 Chiliagon

10,000 Myriagon


Sides Prefix

20 Icosikai...

30 Triacontakai...

40 Tetracontakai...

50 Pentacontakai...60 Hexacontakai...

70 Heptacontakai...

+ Sides Suffix

+1 ...henagon

+2 ...digon

+3 ...trigon

+4 ...tetragon

+5 ...pentagon

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Basic Formulas80 Octacontakai...

90 Enneacontakai...

+6 ...hexagon

+7 ...heptagon

+8 ...octagon

+9 ...enneagon


28 sided polygon - Icosikaioctagon


pi= = 3.141592...)

Area Formulas

Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a 2

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b1 + b2)

circle = pi r2

ellipse = pi r1 r2

triangle=one half times the base length times theheight of the triangle

equilateral triangle =

cube = a3

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Basic Formulas

rectangular prism = a b c

irregular prism = b h

cylinder = b h = pi r 2 h

pyramid = (1/3) b h

cone = (1/3) b h = 1/3 pi r 2 h

sphere = (4/3) pi r 3

ellipsoid = (4/3) pi r1 r2 r3

Surface Area of a Cylinder = 2pir2 + 2pir h

Surface Area of a Sphere = 4pir2

(r is radius of circle)

Surface Area of Any Prism

(b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

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Basic FormulasSurface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)

Surface Area of a Cube = 6 a 2

(a is the length of the side of each edge of the cube)

CIRCLE.

a circle

Definition: A circle is the locus of all points equidistant from a central point.

Definitions Related to Circles

arc: a curved line that is part of the circumference of a circle

chord: a line segment within a circle that touches 2 points on the circle.

circumference: the distance around the circle.

diameter: the longest distance from one end of a circle to the other.

origin: the center of the circle

pi ( ): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle.

radius: distance from center of circle to any point on it.

sector: is like a slice of pie (a circle wedge).

tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.

Diameter = 2 x radius of circleCircumference of Circle = PI x diameter = 2 PI x radius

where PI = = 3.141592...Area of Circle:

area = PI r2

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Basic FormulasLength of a Circular Arc: (with central angle )

if the angle is in degrees, then length = x (PI/180) x r

if the angle is in radians, then length = r x

Area of Circle Sector: (with central angle )

if the angle is in degrees, then area = ( /360)x PI r2

if the angle is in radians, then area = (( /(2PI))x PI r2

Equation of Circle: (Cartesian coordinates)

for a circle with center (j, k) and radius (r): (x-j)^2 + (y-k)^2 = r^2Equation of Circle: (polar coordinates)

for a circle with center (0, 0): r( ) = radius

for a circle with center with polar coordinates: (c, ) and radius a:

r2 - 2cr cos( - ) + c2 = a2Equation of a Circle: (parametric coordinates)

for a circle with origin (j, k) and radius r:x(t) = r cos(t) + j y(t) = r sin(t) + k

square = 4a

rectangle = 2a + 2b

triangle = a + b + c

circle = 2pi r

circle = pid (where d is the diameter)

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Basic FormulasGENERAL MATH FORMULAS

AREA FORMULAS

CIRCLE:r2 (=constant=3.142(approx), and r = radius)

RECTANGLE:

ab (a = width and b = height of rectangle)

SQUARE:

a2 (a = width = height of square, all sides equal)

TRIANGLE:

1/2(bh) (b = base width and h = height of triangle)

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Basic FormulasSURFACE AREA

CYLINDER:

2( r2 + rh) (=constant=3.142(approx), r = radius, h = height) CUBE:

6a2

(a=length of each side of the cube) RECTANGULAR PRISM:

2(ab + ac + bc) (a, b, and c are the lengths of the 3 sides)

SPHERE:

4r2 (=constant=3.142(approx), and r = radius)VOLUME FORMULAS

CUBE:

a3 (a = length = width = height of square, all sides equal)

CONE:

1/3r2h (=constant=3.142(approx), r = radius, h = height) CYLINDER:

r2h (=constant=3.142(approx), r = radius, h = height) PYRAMID:

1/3(bh) (b = base width and h = height of pyramid)

RECTANGULAR PRISM:

abc (a = length, b = width and c = height of rectangular prism )

PERIMETER:

CIRCLE:

d or 2r (=constant=3.142(approx), d = diameter and r =radius)

SQUARE:

4a or a + a + a +a (a = length of each side of square )

RECTANGLE:

2(a + b) or a + b + a + b (a = length, b = height of rectangle)

What is a Polygon?A closed plane figure made up of several line segments that are joined together. The sides do not crosseach other. Exactly two sides meet at every vertex.

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Basic FormulasTypes of PolygonsRegular - all angles are equal and all sides are the same length. Regular polygons are both equiangularand equilateral.Equiangular - all angles are equal.

Equilateral - all sides are the same length.

Convex - a straight line drawn through a convex polygon crosses at most two sides.

Every interior angle is less than 180.

Concave - you can draw at least one straight line through a concave polygon that crossesmore than two sides. At least one interior angle is more than 180.

Polygon Formulas(N = # of sides and S = length from center to a corner)

Area of a regular polygon = (1/2) N sin(360/N) S2

Sum of the interior angles of a polygon = (N - 2) x 180

The number of diagonals in a polygon = 1/2 N(N-3)The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)

Polygon Parts

Side - one of the line segments that make up the polygon.

Vertex - point where two sides meet. Two or more of thesepoints are called vertices.

Diagonal - a line connecting two vertices that isn't a side.

Interior Angle - Angle formed by two adjacent sides insidethe polygon.

Exterior Angle - Angle formed by two adjacent sides

outside the polygon.

Special PolygonsSpecial Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid.


Polygon NamesGenerally accepted names

Sides Name

N N-gon

3 Triangle

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Basic Formulas4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

10 Decagon

12 Dodecagon


Sides Name

9 Nonagon, Enneagon







18 Octadecagon, Octakaidecagon

19 Enneadecagon, Enneakaidecagon

20 Icosagon

30 Triacontagon

40 Tetracontagon

50 Pentacontagon

60 Hexacontagon

70 Heptacontagon80 Octacontagon

90 Enneacontagon


1,000 Chiliagon

10,000 Myriagon


Sides Prefix

20 Icosikai...

30 Triacontakai...

40 Tetracontakai...

50 Pentacontakai...

60 Hexacontakai...

70 Heptacontakai...

80 Octacontakai...

90 Enneacontakai...

+

Sides Suffix

+1 ...henagon

+2 ...digon

+3 ...trigon

+4 ...tetragon

+5 ...pentagon

+6 ...hexagon

+7 ...heptagon

+8 ...octagon

+9 ...enneagon


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Basic Formulas28 sided polygon - Icosikaioctagon


pi= = 3.141592...)

Area FormulasNote: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a 2

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b1 + b2)

circle = pi r2

ellipse = pi r1 r2

Triangle=one half times the base length times theheight of the triangle

equilateral triangle =

cube = a 3

rectangular prism = a b c

irregular prism = b h

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Basic Formulas

cylinder = b h = pi r 2 h

pyramid = (1/3) b h

cone = (1/3) b h = 1/3 pi r 2 h

sphere = (4/3) pi r 3

ellipsoid = (4/3) pi r1 r2 r3

Surface Area of a Cylinder = 2pir2 + 2pir h

Surface Area of a Sphere = 4pir2

(r is radius of circle)

Surface Area of Any Prism

(b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)

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Basic Formulas

Surface Area of a Cube = 6 a 2

(a is the length of the side of each edge of the cube)

CIRCLE.

a circle

Definition: A circle is the locus of all points equidistant from a central point.

Definitions Related to Circles

arc: a curved line that is part of the circumference of a circle

chord: a line segment within a circle that touches 2 points on the circle.

circumference: the distance around the circle.

diameter: the longest distance from one end of a circle to the other.

origin: the center of the circle

pi ( ): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle.

radius: distance from center of circle to any point on it.

sector: is like a slice of pie (a circle wedge).

tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.

Diameter = 2 x radius of circleCircumference of Circle = PI x diameter = 2 PI x radius

where PI = = 3.141592...Area of Circle:

area = PI r2

Length of a Circular Arc: (with central angle )

if the angle is in degrees, then length = x (PI/180) x r

if the angle is in radians, then length = r x

Area of Circle Sector: (with central angle )

if the angle is in degrees, then area = ( /360)x PI r2

if the angle is in radians, then area = (( /(2PI))x PI r2

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Basic FormulasEquation of Circle: (Cartesian coordinates)

for a circle with center (j, k) and radius (r):

(x-j)^2 + (y-k)^2 = r^2Equation of Circle: (polar coordinates)

for a circle with center (0, 0): r( ) = radius

for a circle with center with polar coordinates: (c, ) and radius a:

r2 - 2cr cos( - ) + c2 = a2Equation of a Circle: (parametric coordinates)

for a circle with origin (j, k) and radius r:x(t) = r cos(t) + j y(t) = r sin(t) + k

square = 4a

rectangle = 2a + 2b

triangle = a + b + c

circle = 2pi r

circle = pid (where d is the diameter)

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Basic Formulas

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