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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(Ï•) + 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

    Alternate names

    sublime triangle

    horizontal isosceles triangle

    scalene trigon

    trigon

    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(Ï•) 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(Ï•))/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(Ï•) 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(Ï•))/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 Ï•^(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 Ï•^(3/2) + (a^4 sqrt(Ï•))/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 Ï•)/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(Ï•/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 Ï• | a Ï• 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(Ï•) | a Ï• | 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 Ï• 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 Ï• + 1) isosceles right triangle | p = (2 + sqrt(2)) a isosceles triangle | p = sqrt(a^2 + 4 h^2) + a Kepler triangle | p = a (sqrt(Ï•) + Ï• + 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 (Ï• - 1) (Ï•^(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 Ï•)/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 Ï• isosceles right triangle | sqrt(2) a isosceles triangle | max(a, sqrt(a^2/4 + h^2)) Kepler triangle | a Ï• 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(Ï•))/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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