AREA aa = is perpendicular to a

CIRCLE Orthocenter - common point of altitude

π 2
A = πr2 = d
4
*angle bisector
C = 2πr = dπ
*sector
1
A = r2θ
2
Arc length, S = rθ
*segment 2
Ba =
b+c
√bcs (s − a)
1 2
A= r (θ - sinθ)
2 s = semiperimeter
Arc length, S = rθ r = perpendicular distance from side to incenter
Incenter = common point of angle bisector
TRIANGLE Bisector = divides the angle or two side into two equal
*SIDES sides
Equilateral or equiangular
= all sides and angles are equal SINE LAW
Isoceles a b c
= =
sinA sinB sinC
=2 sides and angles are equal
Scalene
=No sides are equal

COSINE LAW
1
A = bℎ 2 2 2
2 −1 b +c − a
2 2 2
a =b +c −2 bccosA A=cos ( ¿ )¿
2 bc
*median
2 2 2
2 2 2 −1 a +c − b ¿
b =c + a −2 accosB B=cos ( )¿
2 ac
2 2 2
2 2 2 −1 a +b −c ¿
c =a + b −2 abcosC C=cos ( )¿
2 ab
1
Ma=
2
√ 2 2
2b + 2 c −a
2

Complex Mode - sci cal

M a= |a2 − c ∠ B| -Heron’s formula

a+b+ c
Centroid - common point of medians Semiperimeter, s =
2
Area, A =√ s (s − a)(s − b)(s −c )
*altitude
-SAS theorem
1
Area, A = absin θ
2
-3 angles 1 side
2 AT 2
a sin B sin C
aa = Area, A =
a 2sin A
-triangle Inscribe in a Circle -parallelogram
abc Area, A = bh , right parallelogram 90 degress
Area, A =
4r 1
Area, A = d d sin B , B = angle between d1 and d2
a b c 2 1 2
2 r= + +
sinA sinB sinC

Circumcenter = perpendicular bisector of a triangle

-Trapezoid
tha intersect at a common point
1
Area, A = (b1 +b2 )(ℎ)
2
-Triangle Circumscribe in a Circle
Area, A = rs s=semiperimeter -rhombus
1
A= d d
2 1 2
1
S=
2
√ 2
d1 + d2
2

P = 4s
r = perpendicular distance from side to incenter NOTE:
-Triangle Excribing a circle -ALL SIDE OF RHOMBUS IS EQUAL TO EACH
A = r(s-a) OTHER
-THE TWO DIAGONAL IS PERPEDICULAR TO
EACH OTHER

-cyclic quadrilateral ( circumscribing a circle )

QUADRILATERAL a+b+ c+ d
Semiperimeter, s =
-trapezium (General quadrilateral) 2
A= √(s − a)(s − b)(s − c)( s −d )− abcd cos2 θ A = √ (s − a)(s − b)(s − c)( s −d )

a+b+ c+ d
s=
2 -quadrilateral circumscribe in a circle
a+b c+ d
θ= = A = √ abcd
2 2
A = rs

-square
KITE
Area, A = s2
1
Perimeter, P = 4S A= d d
2 1 2
Diagonal, D = S √ 2
ellipse
A = πab
-rectangle

√
2 2
A=BxH P = 2 π a +b
2
P = 2(L + B)
Diagonal, D = √ L2+ B2
 √
POLYGON E s s −a s −b s−c
tan = tan( )tan( )tan( )tan( )
ns
2 4 2 2 2 2
A= 180 Spherical Defect, D = 360 - (A+B+C)
4 tan( )
n Radius of Earth
1
A = pa R = 6373km = 3959 miles
2
Usually 6400km
A = n * Atriangle
1 degree = 60 minutes of arc
Sum of interior angle
1 minute of arc = 1 nautical mile
Σθ = 180(n - 2)
1 nautical mile = 1 statue mile
One exterior angle
1 statue mile = 5280 ft
360
ß= 1 knot = 1 nautical mile per hour
n
One central angle
360
a=
n
Number of diagonals
n
d= (n −3)
2
-INscribe in a circle
1 2 360
A= n r sin( )
2 n
2 180
A = n a tan( )
n

-CIRCUMscbribe in a circle
2 180
A = n r tan ( )
n Right Spherical Triangle
Radius = Apothem
-Inscribe in a square
n 2 180
A= s tan( )
4 n
NOTE:
Ā = 90 - A
n= number of sides
Ḇ = 90 - B
s= length of side
č = 90 - c
a = apothem
SIN CO OP RULE
r=radius The Sine of any middle part is equal to the product of the
cosines of its opposite parts
p = perimeter
sin a = (cos č )(cos Ā)

SIN TA AD RULE
SPHERICAL TRIANGLE
The Sine of any middle part is equal to the product of the
πr E
2
tangents of its adjacent parts.
SA = sin a = (tan b)(tan Ḇ)
180
3
πr E Oblique Spherical Triangle
V=
540
Spherical excess, E = A+B+C - 180
 1 2 2
V= Aℎ= π r ℎ
3 3
SPHERICAL LUNE

Law of Cosines and Sines SA = 2 r 2 ∅

cos a=cos b cos c +sin b sin c sin A Wedge

cos b=cos a cos c +sin a sin c sin B 2 3
V= r ∅
3
cos c=cos a cos b+sin a sin b sinC
Law of Cosines for angle
CYLINDER
cos A=− cos B cos C+sin B sin C sin a
cos B=−cos A cos C+sin A sin C sin b SA = 2 πr (r +ℎ)

cos C=− cos A cos B+sin A sin B sin c V = πr2h

Law of Sines
sin a sin b sin c TETRAHEDRON
= =

√
sin A sin B sin C 2
H=s
3
SA = s2 √ 3
s3
V=
6 √2

PYRAMID
1
SA = A B + PbS
2
1
VOLUME V= A h
3 B
CUBE/HEXAHEDRON
AB = Area of Base
2
SA = 6s
P = perimeter of the base
V = s3
S = slant height
D = s √3
h = perpendicular to the base

SPHERE Triangular PRISM S

SA = 4πr2 SA = 2AB + Ph
4 3 V = AB h
V= πr
3 P = perimeter of the base
h = height
1 2
V= π ℎ (3 r −ℎ) , 1base
3
Rectangular Prism
1 2 2 2
V = πℎ(3 a + 3 b +ℎ ) , 2bases SA=2 LW +2 LH + 2WH
6
A = 2πrh V =LWH

a and b should be in radius for 1 & 2 base

SPHERICAL CONE
SA = πrl+ π r 2 TRUNCATED PRISM or CYLINDER

l = √ r 2 +ℎ2
 V = (A x B) •C
Angle between two vectors or plane
θ = cos −1 ¿ ¿
V Types of No. Of No. Of No. Of Surface Volume VA = Vector A
Polygon
= faces faces vertices edges area VB = Vector B
3
A s
s √3
2
Tethrahedron Triangle 4 4 6
b
6 √2
3
* Hexahedron square 6 8 12 6s2 s

H s
3
√2
Octahedron triangle 8 6 12 2s
2
√3
3
av
2
Dodecahedron pentagon 12 20 30 20.65s 7.66s3
e
Isosahedron Triangle 20 12 30 5S2√ 3 2.183

LA = Pb * Have

CONE (circular base)

SA = πrB(rB + l)
1 1 2
V= A B ℎ= π r ℎ
3 3
AB = Area of Base
h = perpendicular height from the base
l = Slant height

FRUSTUM
ℎ
V= (A + A + √ A 1 A 2)
3 1 2

PRISMATOID
ℎ
V= (A +4 A m + A 2 )
6 1

VECTOR GEOMETRY / 3D GEOMETRY

Area of parallelogram
A = /A x B/
A = ¿ A/ B/sin θ
Area of Triangle
1
A = / A x B /¿
2
Volume of Parallelipiped
NOTE : the distance to “POINT” to any “LINE” is
ALWAYS PERPENDICULAR