Solid Geometry

Front 1

arithmethic progression:

A n = a-1 + (n minus 1) d.,

Sum is equal to, n over 2 ,times 2a1+(n minus 1).d,

common difference, d2 minus d1.

geometric progression:

A n = A-1,times r . raised to n ,minus 1,

common ratio:,r = a-2 ,over a1,

sum of infinite geometric progression:

sum of IGP = A-1 ,over, 1 minus r.

Back 1

properties of platonic solids:

tetrahedron is equal to ,a squared, times square root of 3,

hexahedron is equal to,6 times ,a squared.

octahedron is equal to, 2-a squared ,times square root of 3,

Front 2

volume of solids:

V= Area of base ,times height,

right circular cylinder:

volume= A-b ,times h , or pie r squared h,

lateral area=2 pie ,r h.

pyramid of the cone:

volume=one third ,A-b, h,

right circular cone:

volume= one third,pie r squared, h,

lateral area= pie r l ,

slant height= squareroot of x squared, + h squared,

Back 2

frustum:

volume= one third, (a.1 + a2+ ,square root of a.1 ,a2 times h,

lateral area= one half of p1+p2 ,times L,

sphere:

surface area= 4 pie r squared.

or , pie D squared,

volume= four third of , pie r cubed,

Front 3

spherical zone,

the area of any zone( one base or two bases)

area= 2 pie r h,

(note: when h=2R, the area of the zone will equal to the total surface area of the sphere which is 4 pie r 2,

Back 3

spherical sector,

volume= two third pie r squared h,

area= pie r times 2h + a,

spherical segment,

area=2 pie r h + pie a squared + pie b squared,

volume of spherical segment of two bases,

volume= one six pie h times 3a squared + 3b squared + h squared,

volume of of one base given,

volume= one third pie h squared 3r- h,

Front 4

spherical wedge and spherical lune,

volume of wedge over theta = four third pie r cube divide 360 degrees.

area of lune= area of sphere over 1 revolution.

area of lune over theta= 4 pie r squared divide 360 degrees.

Back 4

area of triangle,

given the base and altitude,

area = one half bh.

given two sides and included angle,

area = one half ab sine C.

given three sides,

area= squareroot of s, s minus a,s minus b, s minus c.

s= a+b+c over 2.

inscribed circle or incircle,

r=Area of triangle over s.

circumscribed or circumcenter,

r= abc over 4 area of triangle.

Front 5

quadrilateral,

perimeter = a + b + c + d

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

general quadrilateral,

Area = one half of d1, d2 sin theta.

common quadrilaterals,

given diagonals d1 and d2:

area = one half of d1 d2.

given side a and one angle A:

area = a squared sin A.

area of parallelogram:

given diagonals d1 and d2 and included angle,

area = one half d1 d2 sine theta,

given two sides a and b and one angle:

area = ab sine A.

Back 5

trapezoid:

area = a + b over 2. -h

cyclic quadrilaterals:

area = squareroot, s minus a, s minus b, s minus c, s minus d.

s = a+b+c+d over 2.

or ,

d1d2 = ac + bd

Front 6

regular polygons,

area of one segment,

A1 = 1/2 r squared sine theta.

total area,

area = n over 2, r squared sine theta.

perimeter, n x.

angle = 360 divide n.

Back 6

circle:

area = pie r squared.

circumference = 2 pie r, or pie d.

length of arc, S = pie r theta(in degree) over 180 degrees.,

or r theta in radians.

area of the sector = pie r squared theta over 360 degrees.

Front 7

area of circular segment: when s less than 1/2 c.

area= area of sector minus area of triangle.

area = 1/2 r squared ( theta in rad minus sin theta in degrees)

area of circular segment: when s more than 1/2 c.

area = 1/2 r squared (theta in radian + sin theta in degree).

Back 7

radius of circle:

circumscribed in triangle,

r = abc over 4 area of triangle.

circle inscribed.

r = area of triangle over S.

circle inscribed in quadrilateral,

r = area of quadrilateral over s.

area of quadrilateral = squareroot abcd.

semi perimeter, s = a + b + c + d over 2.