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Triangular Laminae

Named laminae

30-60-90 triangle | 3, 4, 5 triangle | equilateral triangle | inverted equilateral triangle | golden triangle | isosceles right triangle | isosceles triangle | Kepler triangle | pennant | right triangle | scalene triangle | triangle (total: 12)

Definitions

Definitions 30-60-90 triangle

Definitions 3, 4, 5 triangle

Definitions Equilateral triangle

Defining inequalities

y>=0 and sqrt(3) a>=2 (sqrt(3) x + y) and x>=0

y>=0 and 12 a>=3 x + 4 y and x>=0

sqrt(3) (a + 3 x)>=3 y and sqrt(3) a + 6 y>=0 and sqrt(3) a>=3 (sqrt(3) x + y)

sqrt(3) (a + 3 x)>=-3 y and sqrt(3) a>=6 y and sqrt(3) a + 3 y>=3 sqrt(3) x

-a/2<=x<=a/2 and 0<=y<=1/2 sqrt(5 + 2 sqrt(5)) a (1 - 2 abs(x/a))

y>=0 and a>=x + y and x>=0

-a/2<=x<=a/2 and 0<=y<=h (1 - 2 abs(x/a))

y>=0 and a>=x/sqrt(ϕ) + y and x>=0

h (a + 2 y)>=a x and a h>=a x + 2 h y and x>=0

y>=0 and a b>=a y + b x and x>=0

Lamina properties

30-60-90 triangle | (0, 0) | (a/2, 0) | (0, (sqrt(3) a)/2)
3, 4, 5 triangle | (0, 3 a) | (0, 0) | (4 a, 0)
equilateral triangle | (0, a/sqrt(3)) | (-a/2, -a/(2 sqrt(3))) | (a/2, -a/(2 sqrt(3)))
inverted equilateral triangle | (0, -a/sqrt(3)) | (a/2, a/(2 sqrt(3))) | (-a/2, a/(2 sqrt(3)))
golden triangle | (-a/2, 0) | (a/2, 0) | (0, 1/2 sqrt(5 + 2 sqrt(5)) a)
isosceles right triangle | (0, a) | (0, 0) | (a, 0)
isosceles triangle | (-a/2, 0) | (a/2, 0) | (0, h)
Kepler triangle | (0, 0) | (sqrt(ϕ) a, 0) | (0, a)
pennant | (0, a/2) | (0, -a/2) | (h, 0)
right triangle | (0, 0) | (a, 0) | (0, b)
scalene triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)
triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

30-60-90 triangle | 3
3, 4, 5 triangle | 3
equilateral triangle | 3
inverted equilateral triangle | 3
golden triangle | 3
isosceles right triangle | 3
isosceles triangle | 3
Kepler triangle | 3
pennant | 3
right triangle | 3
scalene triangle | 3
triangle | 3

30-60-90 triangle | a>0
3, 4, 5 triangle | a>0
equilateral triangle | a>0
inverted equilateral triangle | a>0
golden triangle | a>0
isosceles right triangle | a>0
isosceles triangle | a>0 and h>0
Kepler triangle | a>0
pennant | a>0 and h>0
right triangle | a>0 and b>0
scalene triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b and a!=b!=c
triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

equilateral triangle | r = a/(2 sqrt(3))
inverted equilateral triangle | r = a/(2 sqrt(3))

equilateral triangle | h = a/(2 sqrt(3))
inverted equilateral triangle | h = a/(2 sqrt(3))

30-60-90 triangle | (sqrt(3) a)/2
3, 4, 5 triangle | 3 a
equilateral triangle | (sqrt(3) a)/2
inverted equilateral triangle | (sqrt(3) a)/2
golden triangle | 1/2 sqrt(5 + 2 sqrt(5)) a
isosceles right triangle | a
isosceles triangle | h
Kepler triangle | a
pennant | a
right triangle | b
scalene triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)
triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

30-60-90 triangle | A = (sqrt(3) a^2)/8
3, 4, 5 triangle | A = 6 a^2
equilateral triangle | A = (sqrt(3) a^2)/4
inverted equilateral triangle | A = (sqrt(3) a^2)/4
golden triangle | A = 1/4 sqrt(5 + 2 sqrt(5)) a^2
isosceles right triangle | A = a^2/2
isosceles triangle | A = (a h)/2
Kepler triangle | A = (a^2 sqrt(ϕ))/2
pennant | A = (a h)/2
right triangle | A = (a b)/2
scalene triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

30-60-90 triangle | x^_ = (a/6, a/(2 sqrt(3)))
3, 4, 5 triangle | x^_ = ((4 a)/3, a)
equilateral triangle | x^_ = (0, 0)
inverted equilateral triangle | x^_ = (0, 0)
golden triangle | x^_ = (0, 1/6 sqrt(5 + 2 sqrt(5)) a)
isosceles right triangle | x^_ = (a/3, a/3)
isosceles triangle | x^_ = (0, h/3)
Kepler triangle | x^_ = ((sqrt(ϕ) a)/3, a/3)
pennant | x^_ = (h/3, 0)
right triangle | x^_ = (a/3, b/3)
scalene triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))
triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

Mechanical properties

30-60-90 triangle | J_x invisible comma x = (sqrt(3) a^4)/64
3, 4, 5 triangle | J_x invisible comma x = 9 a^4
equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3))
inverted equilateral triangle | J_x invisible comma x = a^4/(32 sqrt(3))
golden triangle | J_x invisible comma x = 1/96 (5 + 2 sqrt(5))^(3/2) a^4
isosceles right triangle | J_x invisible comma x = a^4/12
isosceles triangle | J_x invisible comma x = (a h^3)/12
Kepler triangle | J_x invisible comma x = (a^4 sqrt(ϕ))/12
pennant | J_x invisible comma x = (a^3 h)/48
right triangle | J_x invisible comma x = (a b^3)/12
scalene triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)
triangle | J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

30-60-90 triangle | J_y invisible comma y = a^4/(64 sqrt(3))
3, 4, 5 triangle | J_y invisible comma y = 16 a^4
equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3))
inverted equilateral triangle | J_y invisible comma y = a^4/(32 sqrt(3))
golden triangle | J_y invisible comma y = 1/96 sqrt(5 + 2 sqrt(5)) a^4
isosceles right triangle | J_y invisible comma y = a^4/12
isosceles triangle | J_y invisible comma y = (a^3 h)/48
Kepler triangle | J_y invisible comma y = 1/12 a^4 ϕ^(3/2)
pennant | J_y invisible comma y = (a h^3)/12
right triangle | J_y invisible comma y = (a^3 b)/12
scalene triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)
triangle | J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

30-60-90 triangle | J_zz = a^4/(16 sqrt(3))
3, 4, 5 triangle | J_zz = 25 a^4
equilateral triangle | J_zz = a^4/(16 sqrt(3))
inverted equilateral triangle | J_zz = a^4/(16 sqrt(3))
golden triangle | J_zz = 1/48 (3 + sqrt(5)) sqrt(5 + 2 sqrt(5)) a^4
isosceles right triangle | J_zz = a^4/6
isosceles triangle | J_zz = 1/48 a h (a^2 + 4 h^2)
Kepler triangle | J_zz = 1/12 a^4 ϕ^(3/2) + (a^4 sqrt(ϕ))/12
pennant | J_zz = 1/48 a h (a^2 + 4 h^2)
right triangle | J_zz = 1/12 a b (a^2 + b^2)
scalene triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))
triangle | J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

30-60-90 triangle | J_x invisible comma y = -a^4/128
3, 4, 5 triangle | J_x invisible comma y = -6 a^4
equilateral triangle | J_x invisible comma y = 0
inverted equilateral triangle | J_x invisible comma y = 0
golden triangle | J_x invisible comma y = 0
isosceles right triangle | J_x invisible comma y = -a^4/24
isosceles triangle | J_x invisible comma y = 0
Kepler triangle | J_x invisible comma y = -(a^4 ϕ)/24
pennant | J_x invisible comma y = 0
right triangle | J_x invisible comma y = -1/24 a^2 b^2
scalene triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)
triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

30-60-90 triangle | r_x = a/(2 sqrt(2))
r_y = a/(2 sqrt(6))
3, 4, 5 triangle | r_x = sqrt(3/2) a
r_y = 2 sqrt(2/3) a
equilateral triangle | r_x = a/(2 sqrt(6))
r_y = a/(2 sqrt(6))
inverted equilateral triangle | r_x = a/(2 sqrt(6))
r_y = a/(2 sqrt(6))
golden triangle | r_x = sqrt(5/24 + sqrt(5)/12) a
r_y = a/(2 sqrt(6))
isosceles right triangle | r_x = a/sqrt(6)
r_y = a/sqrt(6)
isosceles triangle | r_x = h/sqrt(6)
r_y = a/(2 sqrt(6))
Kepler triangle | r_x = a/sqrt(6)
r_y = a sqrt(ϕ/6)
pennant | r_x = a/(2 sqrt(6))
r_y = h/sqrt(6)
right triangle | r_x = b/sqrt(6)
r_y = a/sqrt(6)
scalene triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c)
r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))
triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c)
r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

equilateral triangle | K = (sqrt(3) a^4)/80
inverted equilateral triangle | K = (sqrt(3) a^4)/80
isosceles right triangle | K = a^4 (1/12 - (16 sum_n^∞ coth(1/2 π (2 n - 1))/(2 n - 1)^5)/π^5)

Distance properties

30-60-90 triangle | a/2 | a | (sqrt(3) a)/2
3, 4, 5 triangle | 3 a | 4 a | 5 a
equilateral triangle | a | a | a
inverted equilateral triangle | a | a | a
golden triangle | a | a ϕ | a ϕ
isosceles right triangle | a | a | sqrt(2) a
isosceles triangle | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2)
Kepler triangle | a sqrt(ϕ) | a ϕ | a
pennant | a | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2)
right triangle | a | sqrt(a^2 + b^2) | b
scalene triangle | a | b | c
triangle | a | b | c

30-60-90 triangle | a
3, 4, 5 triangle | 5 a
isosceles right triangle | sqrt(2) a
Kepler triangle | a ϕ
right triangle | sqrt(a^2 + b^2)

30-60-90 triangle | p = 1/2 (3 + sqrt(3)) a
3, 4, 5 triangle | p = 12 a
equilateral triangle | p = 3 a
inverted equilateral triangle | p = 3 a
golden triangle | p = a (2 ϕ + 1)
isosceles right triangle | p = (2 + sqrt(2)) a
isosceles triangle | p = sqrt(a^2 + 4 h^2) + a
Kepler triangle | p = a (sqrt(ϕ) + ϕ + 1)
pennant | p = sqrt(a^2 + 4 h^2) + a
right triangle | p = sqrt(a^2 + b^2) + a + b
scalene triangle | p = a + b + c
triangle | p = a + b + c

30-60-90 triangle | r = 1/4 (sqrt(3) - 1) a
3, 4, 5 triangle | r = a
equilateral triangle | r = a/(2 sqrt(3))
inverted equilateral triangle | r = a/(2 sqrt(3))
golden triangle | r = 1/2 sqrt(5 - 2 sqrt(5)) a
isosceles right triangle | r = a - a/sqrt(2)
isosceles triangle | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h)
Kepler triangle | r = 1/2 a (ϕ - 1) (ϕ^(3/2) - 1)
pennant | r = (a (sqrt(a^2 + 4 h^2) - a))/(4 h)
right triangle | r = 1/2 (-sqrt(a^2 + b^2) + a + b)
scalene triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))
triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

30-60-90 triangle | R = a/2
3, 4, 5 triangle | R = (5 a)/2
equilateral triangle | R = a/sqrt(3)
inverted equilateral triangle | R = a/sqrt(3)
golden triangle | R = sqrt(1/10 (5 + sqrt(5))) a
isosceles right triangle | R = a/sqrt(2)
isosceles triangle | R = 1/8 (a^2/h + 4 h)
Kepler triangle | R = (a ϕ)/2
pennant | R = 1/8 (a^2/h + 4 h)
right triangle | R = 1/2 sqrt(a^2 + b^2)
scalene triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

30-60-90 triangle | a
3, 4, 5 triangle | 5 a
equilateral triangle | a
inverted equilateral triangle | a
golden triangle | a ϕ
isosceles right triangle | sqrt(2) a
isosceles triangle | max(a, sqrt(a^2/4 + h^2))
Kepler triangle | a ϕ
pennant | max(a, sqrt(a^2/4 + h^2))
right triangle | sqrt(a^2 + b^2)
scalene triangle | max(a, b, c)
triangle | max(a, b, c)

30-60-90 triangle | χ = 1
3, 4, 5 triangle | χ = 1
equilateral triangle | χ = 1
inverted equilateral triangle | χ = 1
golden triangle | χ = 1
isosceles right triangle | χ = 1
isosceles triangle | χ = 1
Kepler triangle | χ = 1
pennant | χ = 1
right triangle | χ = 1
scalene triangle | χ = 1
triangle | χ = 1

30-60-90 triangle | s^_ = (a (204 + 36 sqrt(3) + 81 log(3) + 2 (9 + 8 sqrt(3)) log(2 + sqrt(3))))/1440
3, 4, 5 triangle | s^_ = (a (20460 + 9728 log(2) + 5103 log(3)))/22500
equilateral triangle | s^_ = 1/20 a (3 + 3 log(3))
inverted equilateral triangle | s^_ = 1/20 a (3 + 3 log(3))
golden triangle | s^_ = 1/60 a ((5 + 2 sqrt(5)) log(1 + 2/sqrt(5)) + 2 (7 + sqrt(5) + sqrt(5) sinh^(-1)(2)))
isosceles right triangle | s^_ = 1/30 a (2 + 4 sqrt(2) + (4 + sqrt(2)) sinh^(-1)(1))
isosceles triangle | s^_ = (-a^5 + 8 a^3 h^2 + 8 a^4 sqrt(a^2 + 4 h^2) + 2 h^2 ((a^2 + 4 h^2)^(3/2) log((a (sqrt(a^2 + 4 h^2) + a))/(2 h^2) + 1) + 16 a^3 log((sqrt(a^2 + 4 h^2) + a)/a)) + 48 a h^4)/(30 a (a^2 + 4 h^2)^(3/2))
Kepler triangle | s^_ = (8 a (7 sqrt(2) + 3 sqrt(10) + 25 sqrt(1 + sqrt(5)) + 11 sqrt(5 (1 + sqrt(5))) + sqrt(2) (7 + 3 sqrt(5)) log(1/2 (1 + sqrt(5) + sqrt(2 (1 + sqrt(5))))) + sqrt(11 + 5 sqrt(5)) log(9 + 4 sqrt(5) + 2 sqrt(38 + 17 sqrt(5))) + sqrt(199 + 89 sqrt(5)) sinh^(-1)(2)))/(15 (1 + sqrt(5))^(9/2))
right triangle | s^_ = (2 a^5 b + 4 a^3 b^3 + a b^4 (sqrt(a^2 + b^2) + 2 b) + a^4 b sqrt(a^2 + b^2) + (a^2 + b^2)^(3/2) (b^3 log((sqrt(a^2 + b^2) + a)/b) + a^3 log((sqrt(a^2 + b^2) + b)/a)) + 2 a^3 b^3 coth^(-1)((a + b)/sqrt(a^2 + b^2)))/(15 a b (a^2 + b^2)^(3/2))
scalene triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)
triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

3, 4, 5 triangle | A^_ = a^2/2
equilateral triangle | A^_ = a^2/(16 sqrt(3))
inverted equilateral triangle | A^_ = a^2/(16 sqrt(3))
golden triangle | A^_ = 1/48 sqrt(5 + 2 sqrt(5)) a^2
isosceles right triangle | A^_ = a^2/24
isosceles triangle | A^_ = (a h)/24
Kepler triangle | A^_ = (a^2 sqrt(ϕ))/24
pennant | A^_ = (a h)/24
right triangle | A^_ = (a b)/24
scalene triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Common classes

convex laminae | polygonal laminae | triangular laminae

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