Math-Solid

Front 1

SOLID GEOMETRY:,

PROPERTIES O FPLATONIC SOLIDS;

Tetrahedron = a squared-square root of 3,

Hexahedron = 6-a squared,

Octahedron = 2-a squared-squareroot of 3,

Prism and Cylinder;

Volume = Ab-h,

Lateral area = Pr-l, pr is perimeter of right section,

Right circular cylinder;

Volume = Ab-h or pie r squared – h,

Lateral area = 2 pier-h,

Pyramid and cone;

Volume = one third Ab-h,

Right circular cone;

Volume = one third pie r squared- h,

Lateral area = pie r-l,

Slant height = squareroot of x squared plus h squared,

Pyramids;

Properties of pyramid;

Ab over Ay = h squared over y squared,

Regular pyramids;

Area of one lateral face = one half x-l,

Lateral area = PL over 2,

Length of lateral edge, e = squareroot of r squared plus h squared,

Slant height,L = squareroot of r squared plus h squared,

Volume = one third Ab –h, or pie over six , r squared – h sin theta,

Truncated prism;

Volume = area of the right section times h-1 plus h-2 plus h-3 over number oh h,

Frustum;

Volume = one third (A-1 plus A-2 plus squareroot of A-1. A2 )- h,

Lateral area = one half (p-1 plus p2 )-L,

Sphere;

Surface area = 4 pie r squared, or pie D squared,

Volume = four third pie r cubed,

Spherical zone;

Area of the zone = circumference of the great circle times altitude,

Area = 2 pie r –h,

Spherical sector;

Total surface area = 2 pie r- h plus pie a-R plus pie b-R or,

Pie r ( 2h+a+b)

Volume = two third pie r squared-h,

Back 1

Spherical segment:

Total area = 2 pie r –h plus pie a squared plus pie b squared,.

Volume of spherical segment of two bases given = one six. Pie –h (3a squared plus 3b squared plus h squared,.

Volume of spherical segment of one base given = one third pie – h squared ( 3r minus h)

Spherical wedge and spherical lune;

Volume of wedge over theta = four third pie r cubed over 360 degrees,

Area of lune over theta = 4 pie r squared over 360 degrees,.

Similar figures;

Any kind of pairs of corresponding line segments ,x-1,x-2 and y-1 , y-2 have the same ratio = x-1 over x-2 = y-1 over y-2,.

Areas of similar surfaces,A-1 and A-2 have the sam ratio =A-1 over A-2 = (x-1 squared) over (x-2 squared ),

Volume of similar solid V-1 and V-2 have tha same ratio = V-1 over V-2 = (x-1 cubed ) over (x-2 cubed),

Area of triangle :

One base and the altitude = one half b-h,

Two base and included angle = one half a-b sin C, or one half a-c sin B,

Given three sides (heros formula ) = squareroot of s (s minus a),(s minus b), (s minus c )

S=perimeter , = a+b+s over 2.,

c

centers of triangle;

radius of incircle, r = Area of triangle over S,

radius of circumcenter, r = abc over 4 Area of triangle,

quadrilateral :,

perimeter = a+b+c+d,

area = squareroot of , (s minus a), (s minus b), (s minus c), (s minus d) – abcd cos squared theta,.

Area can also be expressed in terms of diagonal d-1 and d-2,

Area = one half d-1. D-2 sine theta,.

Common quadrilaterals;

Rhombus a parallelogram with four equal sides,.

Given diagonals d-1 ad d-2 :,

Area = one half d1-d2,.

Given side a and one angle A = a squared sine theta,.

Area of Parallelogram ;

Given diagonals d1 and d2 and included angle theta = one half d1 –d2 sine theta,.

Given two sides a and b and one angle = ab sine A,.

Trapezoid;

Area = a+b over 2 – h,.

Cyclic quadrilaterals a quadrilateral whose vertices lie on the circumference of a circle:

Area = squareroot of (s minus a), (s minus b), (s minus C ), (s minus d),.

D1-d2 = ac + bd ,

Regular polygons;

Area of one segment = one half r squared sine theta,

Total area = n over 2 ( r squared sine theta,.)

Perimeter = n x.,

Angle = 360 over n,

Front 2

Circle;

Area of circle = pie r squared,

Circumference = 2 pie r, or pie d,.

Length of arc , S = pie r theta over 180 degrees

Area of sector = pie r squared theta over 360 degrees,.

Radius of circle:

Radius of circle circumscribed about a triangle = abc over 4 Area of triangle,

Radius of circle inscribed in a triangle = area of triangle over s,

Circle escribed about a triangle, (excircle)

Radius = area of triangle over ,(s minus a)

Circle cimcumscribed about a uadrilateral;

Radius = squareroot of (ab +cd ), (ac + bd ), (ad +bc ), over 4 area of quadrilateral,.

Area of quadrilateral = squareroot of (s minus a), (s minus b), (s minus c), (s minus d),

Circle inscribed in a quadrilateral , radius =area of quadrilateral over s,.

Area of quadrilateral = squareroot of abc,.

Quadratic formula :

Quadratic equation : Ax squared + Bx + C = 0,

X= -b plus minus squareroot of b squared minus 4 ac over 2a,

Where; B squared minus 4 ac called discriminant

If b squared minus 4 ac = 0,. The roots are equal,

If b squared minus 4 ac greater than 0, the roots are real , un equal,.

If b squared minus 4 ac less than 0,the roots are imaginary,

Properties of roots :

Sum of roots, x1 + x2 = negative B over C,

Product of roots, x1-x2 = c over a,

Binomial theorem : in the expansion of (x+y ) raised to n,

Rth term = n ! over ( n-r+1) ! (r-1 )! Times x raised to (n-r+1) times y (r-1)

Middle term = n over 2 plus 1,.

Arithmetic progression :

A-n = a1 + (n minus 1)-d ,

Sum of term = n over 2 (2 a1 + (n minus 1 )-d,.

D= d2 minus d1,

Geometric progression:

A-n = a1-r raised to (n minus 1)

R=a2 over a1

Sum of term , when r is greater than 1, = a1 (r raised to n minus 1) over r minus 1)

When r is less than, a1 (1-r raised to n) over r minus 1)

Sum of infinite geometric progression = a1 over 1 minus r,

Back 2

Sign of function values;

ALL STUDENT TAKE CALCULUS (POSITIVE VALUE)

SUM OF DIFFERENCE OF TWO ANGLE:,

Sine (x+y) = sine x-cosine y + cosine x sine y),

Sine (x-y ) = sine x – cos y minus cosine x – sin y)

Cosine (x+y ) = cosine x cosine y minus sin x-sin y,

Cosine (x-y ) = cosine x cosine y plus sin x-sin y,

Tan (x +y ) = tan x + tan y over 1 plus tanx-tany,

Tan (x -y ) = tan x + tan y over 1 minus tanx-tany,

Double angle identity :

Sine 2x = 2 sine x-cos x,

Cosine 2x = cos squared x minus sin squared x,.

Tangent squared x = 2 tangent x over 1 minus tangent squared x,.

Power of function:

Sine squared x = 1 minus cosine 2x, over 2,.

Cosine squared x = 1 plus cosine 2x, over 2,.

Tangent squared x = 1 minus cosine 2x, over 1 plus cosine 2x,.

Sine law:

A over sine A = b over sine B

Cosine law:

A squared = b squared + c squared – 2bc cosine A,.

Spherical triangle:

A+B+C =180 DEGREES,.

Area of spherical triangle = pie r squared – E, over 180 degrees,.

E = A+B+ C – 180 Degrees,.

Oblique spherical triangle:

Sine law :

Sine a (small) over sine A(big) = sine b (small) over sine B (big)

Cosine law for sides :

Cos –cos .cos + sin.sin.cos,

Cosine a = cosine b-cosine c + sine b-sine c, cosine A (big)

Cosine law for angles : ,

Cosine A = negative cosine B-cosine C + sine C-sine C, cosine a (small)

Front 3

amplitude= the absolute value,

period = 2 pie over the absolute value. ( in parenthesis),

frequency is closely related to period, it is the reciprocal of the period,.

Back 3

sample:for amplitude:

y=10sine x, answer; 10,

period and frequency sample;

y= sine 3x, answer for frequency is 3, period = 2 pie over 3.,