geometry math team formula

Distance Formula

d=square root (x₂ - x₁)²+(y₂ - y₁)²

Midpoint Formula

M = ((x₁ + x₂)/2 , ((y₁ + y₂))/2

30-60-90 Triangle Formula

1:sqrt3:2

45-45-90

1:1:sqrt2

Pole Trick

c=ab/a+b

Area of a triangle

A=bh/2

Area of a trapezoid

A=1/2 h(b₁+b₂)

Area of a parallelogram

A=bh

Area of a regular polygon

A=1/2 ap

Pizza cuts

(n²+n+2)/2

Distance from (x_1,y_1) to Ax+By+C=0

d=|Ax₁+By₁₊C|/sqrt (A²+B²)

Area of an equilateral triangle

A=(s² sqrt3)/4

Area of a rhombus

A=1/2 d₁ d₂ (d is diagonal)

Area of a circle

A=πr²

Area of a sector

A= arc/360 x πr²

Diagonals from 1 point

n-3

Total diagonals

n(n-3)/2

Area of a triangle given the sides (Herron’s)

s=(a+b+c)/2

A = sqrt s(s-a)(s-b)(s-c)

Distance from (x_1,y_1,z_1) to Ax + By + Cz + D=0

d = |Ax₁+By₁+Cz₁+D|/sqrt A²+B²+C²

Cyclical Quadrilateral (Brahmagupta’s)

s=(a+b+c+d)/2

A= sqrt (s-a)(s-b)(s-c)(s-d)

Space Diagonal

sqrt L²+W²+H²

Area of a triangle given coordinates of points

1/2|sum of downhill diagonal products-sum of uphill diagonal products|

Volume of a sphere

V=4/3 πr³

Volume of a prism

V=Bh

Volume of a cylinder

V=πr²h

Volume of a pyramid

V=1/3 Bh

Volume of a cone

V=1/3πr²h

Surface Area of a sphere

SA=4πr²

Surface Area of a prism

SA=2B+ph

Surface Area of a cylinder

SA=2πr²+2πrh

Surface Area of a pyramid

SA=B+1/2 pl

Surface Area of a cone

SA= πr²+πrl

Length of a median (a/b/c sides; c contains median)

median = sqrt a²/2 +b²/2 - c²/4

Altitude from a right angle

seg 1/altitude = altitude/seg 2 | hyp/leg = leg/seg adj hyp

Given the angle bisector

a/d = b/e

equation of a circle

(x-h)²+(y-k)²=r²

Area of a circle given perpendicular chords

A=π x ((a²+b²+c²+d²)/4)

Probability

Probability = true/total

Odds

Odds = favorable/unfavorable

Angle of hands on a clock

Angle = ½ |60H-11M|

Sine

Opposite/hypotenuse

Cosine

Adjacent/hypotenuse

Tangent

opposite/adjacent

Inscribed Angle

m<1 = ½ a

Angle formed by chords

m<1 = ½ (a+b) or sum of intercepted arcs

Angle formed by secants

m<1 = ½ (a-b) or difference of intercepted arcs

Pentagonal numbers (nth row)

n(3n-1)/2

Triangular numbers(nth row)

n(n+1)/2

Chords in circles

ab=cd

Secants in circle

a(a+d)=c(c+b)

Tangents in circles

PA=PB <A=<B=90 deg

Secant and tangent in circles

a²=c(c+b)

Hexagonal numbers (nth row)

n(2n-1)

Ceva’s Theorem

aec=bdf

Euler's Law

F+V = E+2

Radius of inscribed circle

r=(sqrt s(s-a)(s-b)(s-c))/s OR r=(2 x area)/perimete

Relationship between midpoints of diagonals of trapezoids

1/2(HG-EF) (which is bigger side minus smaller side)

Pythagorean Triplets

3-4-5

5-12-1

7-24-25

8-15-17

9-40-41

11-60-61

12-35-37

13-84-85

16-63-65

20-21-29

Quadratic formula

ax²+bx+c=0

x=(-b+-sqrtb²-4ac)/2a

Ptolemy’s theorem

ACxDB=(ADxBC)+(ABxDC)

radius of circumscribed circle

r = ABC/4(area)

radius of circle circumscribed in an equilateral triangle and given height

r=2/3h

radius of circle circumscribed in an equilateral triangle and given side

r=(ssqrt3)/3

Volume of a tetrahedron

V=(s³sqrt2)/12

Centroid of a triangle w/ vertices (x_1,y_1) (x_2, y_2) (x_3, y_3)

((x₁+x₂+x₃)/3), (y₁+y₂+y₃)/3))

Perimeter of a triangle formed by tangents

P=2SA

Stewart’s Theorem

x²b+y²a=z(w²+ab)

Circumscribed circle

2r = sin A/a = sin B/b = sin C/c

Volume of a frustum

V = 1/3(B₁+B₂+sqrtB₁B₂)h

Volume of a triangle given SAS

A = ½ absinC

Area of an octagon given radius

A = 2r²sqrt 2

Area of an octagon given side

A=2s²(1+sqrt2)

Height of a triangle given area and one side

h = (2area)/c

height of equilateral triangle

h=qz+pz+rz

Diagonals of a parallelogram

2(AD²+AB²) = AC²+BD²

Lines in a rectangle

a²+b²=c²+d²

Area of an ellipse

A=pi ab

Quadrilateral with perpendicular diagonals

area = ½ AC x BD (product of diagonals)

AB²+CD²=BC²+AD²

Distance between parallel lines

Ax+By=C and Ax+By=D

d = |C-D|/sqrt A²+B²

Length of an angle bisector

length= 1/a+b sqrt ab(a+b+c)(a+b-c)

Surface area of a frustum

SA = pi (R+r)sqrt (R-r)²+h²