Calculating Purple Area.
tanb = 2/1
b = atan(2)°
tanc = ½
c = atan(½)°
d = 180-b
d = (180-atan(2))°
e = 180-d-c
e = 180-(180-atan(2))-atan(½)
e = (atan(2)-atan(½))°
It implies;
(1/sin(atan(2)-...
Let 1 unit be the side length of the green square.
Therefore;
EN = ½ units.
a = (1+2x) units.
a is the diameter of the bigger circle.
b = ½(a)
b = ½(1+2x) units.
b is the radius of the...
R ~ (a+b)
r ~ a
Cross Multiply.
aR = r(a+b)
R/r = (a+b)/a
R/r = 1+(b/a)
And R/r = phi (golden ratio)
Let phi be x.
x-1 = b/a
a/b = 1/(x-1)
Where x (phi) is;
x = ½(1+√(5))...
Calculating Area A Divided By Area B (A/B).
Let a be the two equal lengths.
b = (3-a) units.
c² = 4²+3²
c = 5 units.
Therefore;
Calculating a.
5 ~ (3-a)
4 ~ a
Cross Multiply.
5x = 12-4a
9a = 1...
Let DC = 1 unit.
a = 180-36+30
a = 114°
a is angle BDC.
(1/sin36) = (b/sin114)
b = 1.55421636402 units.
b is BC.
c = 180-60-54
c = 180-114
c = 66°
c is angle BAC.
Therefore;
(1.55421636402/sin6...
Calculating R.
3R+4R+5R = 3*4
12R = 12
R = 1 units.
Calculating r.
tana = 3/4
a = atan(3/4)°
b = ½(a)
b = ½(atan(3/4))°
tan(½(atan(3/4))) = r/c
c = 3r units.
d = 4-1-3r
d = (3-3r) units.
e...
Let the height and base of the brown area be x and y respectively.
Therefore;
½(3+x)(4+y)-42 = ½(xy)
12+3y+4x+xy-84 = xy
3y+4x = 72 --- (1).
3 ~ (3+x)
4 ~ (y+4)
Cross Multiply.
4(...
a² = 2(1)²
a = √(2) units.
a is the diagonal of the inscribed square and also the side length of the two congruent inside bigger regular triangle.
b = ½(a)
b = ½√(2) units.
c²+b² = a²
c² = √(2)²-...
a = 2(20)
a = 40 units.
b = 60+20
b = 80 units.
c = 2(20)
c = 40 units.
d = atan(80/40)+atan(20/40)
d = 90°
d is angle ACB
Area Triangle ABC is;
½*√(80²+40²)*√(40²+20²)
= ½*√(6400+1600)*√(1600...
Let the inscribed green square side length be a.
2b² = a²
b = ½√(2)a units.
c² = 2(4)²
c = 4√(2) units.
c is the diagonal of the ascribed bigger square.
It implies;
4+a+½(a) = c...
