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    Polygonal Laminae

    Named laminae

    30-60-90 triangle | 3, 4, 5 triangle | diamond | equilateral triangle | inverted equilateral triangle | golden rectangle | golden rhombus | golden triangle | half rectangle | half square | hexagram | isosceles pentagon | isosceles right pentagon | isosceles right triangle | isosceles trapezoid | isosceles triangle | Kepler triangle | kite | lozenge | monokite | ... (total: 58)

    Definitions

    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

    abs(x)/a + abs(y)/b<=1

    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 -(a ϕ)/2<=y<=(a ϕ)/2

    2 (1 + sqrt(5)) a + sqrt(2 (5 + sqrt(5))) (2 x + sqrt(5) y + y)>=0 and 2 (1 + sqrt(5)) a + sqrt(2 (5 + sqrt(5))) (-2 x + sqrt(5) y + y)>=0 and 2 sqrt(5 (5 + sqrt(5))) a>=5 sqrt(2) (2 x + sqrt(5) y + y) and 2 (1 + sqrt(5)) a>=sqrt(2 (5 + sqrt(5))) (-2 x + sqrt(5) y + y)

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

    -a/2<=x<=a/2 and 0<=y<=b/2

    -a/2<=x<=a/2 and 0<=y<=a/2

    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) diamond | (-a, 0) | (0, -b) | (a, 0) | (0, b) 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 rectangle | (a/2, -(ϕ a)/2) | (a/2, (ϕ a)/2) | (-a/2, (ϕ a)/2) | (-a/2, -(ϕ a)/2) golden rhombus | (-a/(2 sqrt(5/8 - sqrt(5)/8)), 0) | (0, -a/(2 sqrt(5/8 + sqrt(5)/8))) | (a/(2 sqrt(5/8 - sqrt(5)/8)), 0) | (0, a/(2 sqrt(5/8 + sqrt(5)/8))) golden triangle | (-a/2, 0) | (a/2, 0) | (0, 1/2 sqrt(5 + 2 sqrt(5)) a) half rectangle | (-a/2, 0) | (a/2, 0) | (a/2, b/2) | (-a/2, b/2) half square | (-a/2, 0) | (a/2, 0) | (a/2, a/2) | (-a/2, a/2) hexagram | (0, a/sqrt(3)) | (-a/6, a/(2 sqrt(3))) | (-a/2, a/(2 sqrt(3))) | (-a/3, 0) | (-a/2, -a/(2 sqrt(3))) | (-a/6, -a/(2 sqrt(3))) | (0, -a/sqrt(3)) | (a/6, -a/(2 sqrt(3))) | (a/2, -a/(2 sqrt(3))) | (a/3, 0) | (a/2, a/(2 sqrt(3))) | (a/6, a/(2 sqrt(3))) isosceles pentagon | (-a/2, 0) | (a/2, 0) | (w, d) | (0, h) | (-w, d) isosceles right pentagon | (-a/2, 0) | (a/2, 0) | (a/2, b) | (0, h) | (-a/2, b) isosceles right triangle | (0, a) | (0, 0) | (a, 0) isosceles trapezoid | (-b/2, 0) | (b/2, 0) | (a/2, h) | (-a/2, h) isosceles triangle | (-a/2, 0) | (a/2, 0) | (0, h) Kepler triangle | (0, 0) | (sqrt(ϕ) a, 0) | (0, a) kite | (-sqrt(a^2 - h^2), 0) | (0, -h) | (sqrt(b^2 - h^2), 0) | (0, h) lozenge | (0, 0) | (a, 0) | ((1 + 1/sqrt(2)) a, a/sqrt(2)) | (a/sqrt(2), a/sqrt(2)) monokite | (0, 0) | ((sqrt(3) a)/2, (3 a)/2) | (0, 2 a) | (-(sqrt(3) a)/2, (3 a)/2) octagram | (0, sqrt(1 - 1/sqrt(2)) a) | (-1/2 sqrt(5 - 7/sqrt(2)) a, 1/2 sqrt(1 - 1/sqrt(2)) a) | (root of 1 - 8 x^2 + 8 x^4 near x = -0.382683 a, 1/2 sqrt(2 - sqrt(2)) a) | (-1/2 sqrt(1 - 1/sqrt(2)) a, 1/2 sqrt(5 - 7/sqrt(2)) a) | (-sqrt(1 - 1/sqrt(2)) a, 0) | (-1/2 sqrt(1 - 1/sqrt(2)) a, -1/2 sqrt(5 - 7/sqrt(2)) a) | (root of 1 - 8 x^2 + 8 x^4 near x = -0.382683 a, root of 1 - 8 x^2 + 8 x^4 near x = -0.382683 a) | (-1/2 sqrt(5 - 7/sqrt(2)) a, -1/2 sqrt(1 - 1/sqrt(2)) a) | (0, -sqrt(1 - 1/sqrt(2)) a) | (1/2 sqrt(5 - 7/sqrt(2)) a, -1/2 sqrt(1 - 1/sqrt(2)) a) | (1/2 sqrt(2 - sqrt(2)) a, root of 1 - 8 x^2 + 8 x^4 near x = -0.382683 a) | (1/2 sqrt(1 - 1/sqrt(2)) a, -1/2 sqrt(5 - 7/sqrt(2)) a) | (sqrt(1 - 1/sqrt(2)) a, 0) | (1/2 sqrt(1 - 1/sqrt(2)) a, 1/2 sqrt(5 - 7/sqrt(2)) a) | (1/2 sqrt(2 - sqrt(2)) a, 1/2 sqrt(2 - sqrt(2)) a) | (1/2 sqrt(5 - 7/sqrt(2)) a, 1/2 sqrt(1 - 1/sqrt(2)) a) parallelogram | (0, 0) | (b, 0) | (cos(A) a + b, sin(A) a) | (cos(A) a, sin(A) a) pennant | (0, a/2) | (0, -a/2) | (h, 0) pentagram | (a/2, 1/2 sqrt(1 - 2/sqrt(5)) a) | (1/2 (-2 + sqrt(5)) a, 1/2 sqrt(1 - 2/sqrt(5)) a) | (0, root of 1 - 5 x^2 + 5 x^4 near x = 0.525731 a) | ((1 - sqrt(5)/2) a, 1/2 sqrt(1 - 2/sqrt(5)) a) | (-a/2, 1/2 sqrt(1 - 2/sqrt(5)) a) | (1/4 (-3 + sqrt(5)) a, -a/sqrt(130 + 58 sqrt(5))) | (1/4 (1 - sqrt(5)) a, root of 1 - 20 x^2 + 80 x^4 near x = -0.425325 a) | (0, root of 1 - 25 x^2 + 5 x^4 near x = -0.200811 a) | (1/4 (-1 + sqrt(5)) a, root of 1 - 20 x^2 + 80 x^4 near x = -0.425325 a) | (1/4 (3 - sqrt(5)) a, -a/sqrt(130 + 58 sqrt(5))) quarter rectangle | (0, 0) | (a/2, 0) | (a/2, b/2) | (0, b/2) quarter square | (0, 0) | (a/2, 0) | (a/2, a/2) | (0, a/2) rectangle | (-a/2, -b/2) | (a/2, -b/2) | (a/2, b/2) | (-a/2, b/2) regular decagon | (1/2 (1 + sqrt(5)) a, 0) | (1/4 (3 + sqrt(5)) a, sqrt(5/8 + sqrt(5)/8) a) | (a/2, 1/2 sqrt(5 + 2 sqrt(5)) a) | (-a/2, 1/2 sqrt(5 + 2 sqrt(5)) a) | (1/4 (-3 - sqrt(5)) a, sqrt(5/8 + sqrt(5)/8) a) | (1/2 (-1 - sqrt(5)) a, 0) | (1/4 (-3 - sqrt(5)) a, -1/2 sqrt(1/2 (5 + sqrt(5))) a) | (-a/2, -1/2 sqrt(5 + 2 sqrt(5)) a) | (a/2, -1/2 sqrt(5 + 2 sqrt(5)) a) | (1/4 (3 + sqrt(5)) a, -1/2 sqrt(1/2 (5 + sqrt(5))) a) regular diamond | (-a, 0) | (0, -a) | (a, 0) | (0, a) regular heptagon | (1/2 cos(π/14) csc(π/7) a, -1/2 csc(π/7) sin(π/14) a) | (1/2 cos((3 π)/14) csc(π/7) a, 1/2 csc(π/7) sin((3 π)/14) a) | (0, 1/2 csc(π/7) a) | (-1/2 cos((3 π)/14) csc(π/7) a, 1/2 csc(π/7) sin((3 π)/14) a) | (-1/2 cos(π/14) csc(π/7) a, -1/2 csc(π/7) sin(π/14) a) | (-a/2, -1/2 cot(π/7) a) | (a/2, -1/2 cot(π/7) a) regular hexagon | (a, 0) | (a/2, (sqrt(3) a)/2) | (-a/2, (sqrt(3) a)/2) | (-a, 0) | (-a/2, -(sqrt(3) a)/2) | (a/2, -(sqrt(3) a)/2) regular nonagon | (-1/2 cos(π/18) csc(π/9) a, 1/2 csc(π/9) sin(π/18) a) | (-1/4 sqrt(3) csc(π/9) a, -1/4 csc(π/9) a) | (-a/2, -1/2 cot(π/9) a) | (a/2, -1/2 cot(π/9) a) | (1/4 sqrt(3) csc(π/9) a, -1/4 csc(π/9) a) | (1/2 cos(π/18) csc(π/9) a, 1/2 csc(π/9) sin(π/18) a) | (1/2 csc(π/9) sin((2 π)/9) a, 1/2 cos((2 π)/9) csc(π/9) a) | (0, 1/2 csc(π/9) a) | (-1/2 csc(π/9) sin((2 π)/9) a, 1/2 cos((2 π)/9) csc(π/9) a) regular octagon | (1/2 cot(π/8) a, a/2) | (a/2, 1/2 cot(π/8) a) | (-a/2, 1/2 cot(π/8) a) | (-1/2 cot(π/8) a, a/2) | (-1/2 cot(π/8) a, -a/2) | (-a/2, -1/2 cot(π/8) a) | (a/2, -1/2 cot(π/8) a) | (1/2 cot(π/8) a, -a/2) regular pentagon | (1/4 (1 + sqrt(5)) a, root of 1 - 20 x^2 + 80 x^4 near x = 0.262866 a) | (0, sqrt(1/10 (5 + sqrt(5))) a) | (1/4 (-1 - sqrt(5)) a, root of 1 - 20 x^2 + 80 x^4 near x = 0.262866 a) | (-a/2, -1/2 sqrt(1 + 2/sqrt(5)) a) | (a/2, -1/2 sqrt(1 + 2/sqrt(5)) a) rhombus | (-cos(θ) a, 0) | (0, -sin(θ) a) | (cos(θ) a, 0) | (0, sin(θ) a) right trapezoid | (0, 0) | (a, 0) | (a, h_2) | (0, h_1) 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) square | (-a/2, -a/2) | (a/2, -a/2) | (a/2, a/2) | (-a/2, a/2) star of Lakshmi | (0, a/sqrt(2)) | ((1/2 - 1/sqrt(2)) a, a/2) | (-a/2, a/2) | (-a/2, (-1/2 + 1/sqrt(2)) a) | (-a/sqrt(2), 0) | (-a/2, (1/2 - 1/sqrt(2)) a) | (-a/2, -a/2) | ((1/2 - 1/sqrt(2)) a, -a/2) | (0, -a/sqrt(2)) | ((-1/2 + 1/sqrt(2)) a, -a/2) | (a/2, -a/2) | (a/2, (1/2 - 1/sqrt(2)) a) | (a/sqrt(2), 0) | (a/2, (-1/2 + 1/sqrt(2)) a) | (a/2, a/2) | ((-1/2 + 1/sqrt(2)) a, a/2) trapezoid | (0, 0) | (b, 0) | ((a^2 - b^2 - c^2 + d^2)/(2 (a - b)), sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (-a + b))) | (-(a^2 - 2 a b + b^2 + c^2 - d^2)/(2 (a - b)), sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (-a + b))) 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) hat polykite | (-a, -sqrt(3) a) | (a, -sqrt(3) a) | ((3 a)/2, -(sqrt(3) a)/2) | (3 a, -sqrt(3) a) | ((9 a)/2, -(sqrt(3) a)/2) | (4 a, 0) | (3 a, 0) | (3 a, sqrt(3) a) | ((3 a)/2, (3 sqrt(3) a)/2) | (a, sqrt(3) a) | (0, sqrt(3) a) | (0, 0) | (-(3 a)/2, -(sqrt(3) a)/2) inverted isosceles trapezoid | (-b/2, 0) | (b/2, 0) | (a/2, sqrt(-1/4 (-a + b)^2 + c^2)) | (-a/2, sqrt(-1/4 (-a + b)^2 + c^2)) inverted T-beam | (b/2, 0) | (b/2, d - h) | (t/2, d - h) | (t/2, d) | (-t/2, d) | (-t/2, d - h) | (-b/2, d - h) | (-b/2, 0) rectangular cross beam | (-b/2, -s/2) | (-t/2, -s/2) | (-t/2, -d/2) | (t/2, -d/2) | (t/2, -s/2) | (b/2, -s/2) | (b/2, s/2) | (t/2, s/2) | (t/2, d/2) | (-t/2, d/2) | (-t/2, s/2) | (-b/2, s/2) rectangular L-beam | (0, 0) | (a, 0) | (a, t) | (t, t) | (t, b) | (0, b) square channel | (0, -d/2) | (b, -d/2) | (b, -h/2) | (t, -h/2) | (t, h/2) | (b, h/2) | (b, d/2) | (0, d/2) square cross | (a/2, a/2) | (a/2, h/2) | (-a/2, h/2) | (-a/2, a/2) | (-h/2, a/2) | (-h/2, -a/2) | (-a/2, -a/2) | (-a/2, -h/2) | (a/2, -h/2) | (a/2, -a/2) | (h/2, -a/2) | (h/2, a/2) square I-beam | (-b/2, -d/2) | (b/2, -d/2) | (b/2, -h/2) | (t/2, -h/2) | (t/2, h/2) | (b/2, h/2) | (b/2, d/2) | (-b/2, d/2) | (-b/2, h/2) | (-t/2, h/2) | (-t/2, -h/2) | (-b/2, -h/2) square L-beam | (0, 0) | (a, 0) | (a, t) | (t, t) | (t, a) | (0, a) square T-beam | (-t/2, 0) | (t/2, 0) | (t/2, h) | (b/2, h) | (b/2, d) | (-b/2, d) | (-b/2, h) | (-t/2, h) swastika | (a/2, t/2) | (t/2, t/2) | (t/2, 1/2 (a - 2 t)) | (a/2, 1/2 (a - 2 t)) | (a/2, a/2) | (-t/2, a/2) | (-t/2, t/2) | (1/2 (-a + 2 t), t/2) | (1/2 (-a + 2 t), a/2) | (-a/2, a/2) | (-a/2, -t/2) | (-t/2, -t/2) | (-t/2, 1/2 (-a + 2 t)) | (-a/2, 1/2 (-a + 2 t)) | (-a/2, -a/2) | (t/2, -a/2) | (t/2, -t/2) | (1/2 (a - 2 t), -t/2) | (1/2 (a - 2 t), -a/2) | (a/2, -a/2) tapered I-beam | (-b/2, -d/2) | (b/2, -d/2) | (b/2, -h/2) | (t/2, -l/2) | (t/2, l/2) | (b/2, h/2) | (b/2, d/2) | (-b/2, d/2) | (-b/2, h/2) | (-t/2, l/2) | (-t/2, -l/2) | (-b/2, -h/2) tilted square cross | (0, a/sqrt(2)) | ((a - h)/(2 sqrt(2)), (a + h)/(2 sqrt(2))) | (-(a + h)/(2 sqrt(2)), (-a + h)/(2 sqrt(2))) | (-a/sqrt(2), 0) | (-(a + h)/(2 sqrt(2)), (a - h)/(2 sqrt(2))) | ((a - h)/(2 sqrt(2)), -(a + h)/(2 sqrt(2))) | (0, -a/sqrt(2)) | ((-a + h)/(2 sqrt(2)), -(a + h)/(2 sqrt(2))) | ((a + h)/(2 sqrt(2)), (a - h)/(2 sqrt(2))) | (a/sqrt(2), 0) | ((a + h)/(2 sqrt(2)), (-a + h)/(2 sqrt(2))) | ((-a + h)/(2 sqrt(2)), (a + h)/(2 sqrt(2))) uneven square I-beam | (-a/2, 0) | (a/2, 0) | (a/2, (d - h)/2) | (t/2, (d - h)/2) | (t/2, d + 1/2 (-d + h)) | (b/2, d + 1/2 (-d + h)) | (b/2, d) | (-b/2, d) | (-b/2, d + 1/2 (-d + h)) | (-t/2, d + 1/2 (-d + h)) | (-t/2, (d - h)/2) | (-a/2, (d - h)/2) Mitsubishi symbol-like lamina | (-(sqrt(3) a)/2, -a/2) | (-a/(2 sqrt(3)), -a/2) | (a/(2 sqrt(3)), a/2) | (0, a) | (-a/(2 sqrt(3)), a/2) | (a/(2 sqrt(3)), -a/2) | ((sqrt(3) a)/2, -a/2) | (a/sqrt(3), 0) | (-a/sqrt(3), 0) Christmas tree with straight branches lamina | (-(sqrt(3) a)/2, a/2) | (-(23 a)/(34 sqrt(3)), a/2) | (-(13 sqrt(3) a)/17, -(19 a)/17) | (-(69 sqrt(3) a)/170, -(19 a)/17) | (-(19 sqrt(3) a)/20, -(11 a)/4) | (-(3 a)/11, -(11 a)/4) | (-(3 a)/11, -(15 a)/4) | ((3 a)/11, -(15 a)/4) | ((3 a)/11, -(11 a)/4) | ((19 sqrt(3) a)/20, -(11 a)/4) | ((69 sqrt(3) a)/170, -(19 a)/17) | ((13 sqrt(3) a)/17, -(19 a)/17) | ((23 a)/(34 sqrt(3)), a/2) | ((sqrt(3) a)/2, a/2) | (0, 2 a)

    30-60-90 triangle | 3 3, 4, 5 triangle | 3 diamond | 4 equilateral triangle | 3 inverted equilateral triangle | 3 golden rectangle | 4 golden rhombus | 4 golden triangle | 3 half rectangle | 4 half square | 4 hexagram | 12 isosceles pentagon | 5 isosceles right pentagon | 5 isosceles right triangle | 3 isosceles trapezoid | 4 isosceles triangle | 3 Kepler triangle | 3 kite | 4 lozenge | 4 monokite | 4 octagram | 16 parallelogram | 4 pennant | 3 pentagram | 10 quarter rectangle | 4 quarter square | 4 rectangle | 4 regular decagon | 10 regular diamond | 4 regular heptagon | 7 regular hexagon | 6 regular nonagon | 9 regular octagon | 8 regular pentagon | 5 rhombus | 4 right trapezoid | 4 right triangle | 3 scalene triangle | 3 square | 4 star of Lakshmi | 16 trapezoid | 4 triangle | 3 hat polykite | 13 inverted isosceles trapezoid | 4 inverted T-beam | 8 rectangular cross beam | 12 rectangular L-beam | 6 square channel | 8 square cross | 12 square I-beam | 12 square L-beam | 6 square T-beam | 8 swastika | 20 tapered I-beam | 12 tilted square cross | 12 uneven square I-beam | 12 Mitsubishi symbol-like lamina | 9 Christmas tree with straight branches lamina | 15

    30-60-90 triangle | a>0 3, 4, 5 triangle | a>0 diamond | a>0 and b>0 equilateral triangle | a>0 inverted equilateral triangle | a>0 golden rectangle | a>0 golden rhombus | a>0 golden triangle | a>0 half rectangle | a>0 and b>0 half square | a>0 hexagram | a>0 isosceles pentagon | 00 and 00 isosceles trapezoid | 00 isosceles triangle | a>0 and h>0 Kepler triangle | a>0 kite | 00 monokite | a>0 octagram | a>0 parallelogram | a>0 and b>0 and 00 and h>0 pentagram | a>0 quarter rectangle | a>0 and b>0 quarter square | a>0 rectangle | a>0 and b>0 regular decagon | a>0 regular diamond | a>0 regular heptagon | a>0 regular hexagon | a>0 regular nonagon | a>0 regular octagon | a>0 regular pentagon | a>0 rhombus | a>0 and 0<θ<π/2 right trapezoid | a>0 and h_1>0 and h_2>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 square | a>0 star of Lakshmi | a>0 trapezoid | a>0 and b>0 and c>0 and d>0 and (a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)>0 triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b hat polykite | a>0 inverted isosceles trapezoid | 0(a - b)/2 inverted T-beam | 00 Christmas tree with straight branches lamina | a>0

    diamond | 2 a | 2 b golden rectangle | a sqrt(ϕ^2 + 1) | a sqrt(ϕ^2 + 1) golden rhombus | a/sqrt(5/8 - sqrt(5)/8) | a/sqrt(5/8 + sqrt(5)/8) isosceles pentagon | sqrt(1/4 (a + 2 w)^2 + d^2) | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) | sqrt((a/2 + w)^2 + d^2) | 2 w isosceles right pentagon | sqrt(a^2 + b^2) | sqrt(a^2/4 + h^2) | sqrt(a^2/4 + h^2) | sqrt(a^2 + b^2) | a isosceles trapezoid | 1/2 sqrt((a + b)^2 + 4 h^2) | 1/2 sqrt((a + b)^2 + 4 h^2) kite | sqrt(a^2 - h^2) + sqrt(b^2 - h^2) | 2 h lozenge | sqrt(2 + sqrt(2)) a | sqrt(2 - sqrt(2)) a monokite | 2 a | sqrt(3) a parallelogram | sqrt(a^2 - 2 a b cos(A) + b^2) | sqrt(a^2 + 2 a b cos(A) + b^2) rectangle | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) regular decagon | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | sqrt(1/2 (5 + sqrt(5))) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | 1/2 (3 + sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | sqrt(5 + 2 sqrt(5)) a | (1 + sqrt(5)) a | (1 + sqrt(5)) a | (1 + sqrt(5)) a | (1 + sqrt(5)) a | (1 + sqrt(5)) a regular diamond | 2 a | 2 a regular heptagon | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - x^2 - 2 x + 1 near x = 1.80194 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 | a root of x^3 - 2 x^2 - x + 1 near x = 2.24698 regular hexagon | sqrt(3) a | sqrt(3) a | sqrt(3) a | sqrt(3) a | sqrt(3) a | sqrt(3) a | 2 a | 2 a | 2 a regular nonagon | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x - 1 near x = 1.87939 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 3 near x = 2.53209 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 | a root of x^3 - 3 x^2 + 1 near x = 2.87939 regular octagon | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | sqrt(2 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | (1 + sqrt(2)) a | sqrt(2 (2 + sqrt(2))) a | sqrt(2 (2 + sqrt(2))) a | sqrt(2 (2 + sqrt(2))) a | sqrt(2 (2 + sqrt(2))) a regular pentagon | a ϕ | a ϕ | a ϕ | a ϕ | a ϕ rhombus | 2 a cos(θ) | 2 a sin(θ) right trapezoid | sqrt(a^2 + h_1^2) | sqrt(a^2 + h_2^2) square | sqrt(2) a | sqrt(2) a trapezoid | sqrt((a^2 (-b) + a b^2 - a c^2 + b d^2)/(b - a)) | sqrt((a^2 (-b) + a b^2 - a d^2 + b c^2)/(b - a)) inverted isosceles trapezoid | sqrt(a b + c^2) | sqrt(a b + c^2)

    equilateral triangle | r = a/(2 sqrt(3)) inverted equilateral triangle | r = a/(2 sqrt(3)) regular decagon | r = 1/2 sqrt(5 + 2 sqrt(5)) a regular heptagon | r = 1/2 a cot(π/7) regular hexagon | r = (sqrt(3) a)/2 regular nonagon | r = 1/2 a cot(π/9) regular octagon | r = 1/2 (1 + sqrt(2)) a regular pentagon | r = 1/10 sqrt(25 + 10 sqrt(5)) a square | r = a/2

    equilateral triangle | h = a/(2 sqrt(3)) inverted equilateral triangle | h = a/(2 sqrt(3)) regular decagon | h = 1/2 (1 + sqrt(5) - sqrt(5 + 2 sqrt(5))) a regular heptagon | h = 1/2 a tan(π/14) regular hexagon | h = -1/2 (sqrt(3) - 2) a regular nonagon | h = 1/2 a tan(π/18) regular octagon | h = 1/2 (-1 - sqrt(2) + sqrt(2 (2 + sqrt(2)))) a regular pentagon | h = 1/2 sqrt(1 - 2/sqrt(5)) a square | h = 1/2 (sqrt(2) - 1) a

    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 rectangle | a ϕ golden triangle | 1/2 sqrt(5 + 2 sqrt(5)) a half rectangle | b/2 half square | a/2 isosceles pentagon | h isosceles right pentagon | h isosceles right triangle | a isosceles trapezoid | h isosceles triangle | h Kepler triangle | a monokite | 2 a parallelogram | a sin(A) pennant | a quarter rectangle | b/2 quarter square | a/2 rectangle | b regular pentagon | 1/2 sqrt(5 + 2 sqrt(5)) a right triangle | b scalene triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c) square | a trapezoid | sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d))/(2 (b - a)) triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c) inverted isosceles trapezoid | sqrt(c^2 - 1/4 (b - a)^2) Mitsubishi symbol-like lamina | (3 a)/2

    30-60-90 triangle | A = (sqrt(3) a^2)/8 3, 4, 5 triangle | A = 6 a^2 diamond | A = 2 a b equilateral triangle | A = (sqrt(3) a^2)/4 inverted equilateral triangle | A = (sqrt(3) a^2)/4 golden rectangle | A = a^2 ϕ golden rhombus | A = (2 a^2)/sqrt(5) golden triangle | A = 1/4 sqrt(5 + 2 sqrt(5)) a^2 half rectangle | A = (a b)/2 half square | A = a^2/2 hexagram | A = a^2/sqrt(3) isosceles pentagon | A = (a d)/2 + h w isosceles right pentagon | A = 1/2 a (b + h) isosceles right triangle | A = a^2/2 isosceles trapezoid | A = 1/2 h (a + b) isosceles triangle | A = (a h)/2 Kepler triangle | A = (a^2 sqrt(ϕ))/2 kite | A = h (sqrt(a^2 - h^2) + sqrt(b^2 - h^2)) lozenge | A = a^2/sqrt(2) monokite | A = sqrt(3) a^2 octagram | A = (6 sqrt(2) - 8) a^2 parallelogram | A = a b sin(A) pennant | A = (a h)/2 pentagram | A = 1/4 sqrt(650 - 290 sqrt(5)) a^2 quarter rectangle | A = (a b)/4 quarter square | A = a^2/4 rectangle | A = a b regular decagon | A = 5/2 sqrt(5 + 2 sqrt(5)) a^2 regular diamond | A = 2 a^2 regular heptagon | A = 7/4 a^2 cot(π/7) regular hexagon | A = (3 sqrt(3) a^2)/2 regular nonagon | A = 9/4 a^2 cot(π/9) regular octagon | A = 2 (1 + sqrt(2)) a^2 regular pentagon | A = 5/4 sqrt(1 + 2/sqrt(5)) a^2 rhombus | A = a^2 sin(2 θ) right trapezoid | A = 1/2 a (h_1 + 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)) square | A = a^2 star of Lakshmi | A = 2 (2 - sqrt(2)) a^2 trapezoid | A = ((a + b) sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)))/(4 (b - a)) triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) hat polykite | A = 8 sqrt(3) a^2 inverted isosceles trapezoid | A = 1/4 (a + b) sqrt(4 c^2 - (b - a)^2) inverted T-beam | A = b (d - h) + h t rectangular cross beam | A = b s + t (d - s) rectangular L-beam | A = t (a + b - t) square channel | A = b (d - h) + h t square cross | A = a (2 h - a) square I-beam | A = b (d - h) + h t square L-beam | A = t (2 a - t) square T-beam | A = b (d - h) + h t swastika | A = t (4 a - 3 t) tapered I-beam | A = 1/2 (b (2 d - h - l) + t (h + l)) tilted square cross | A = a (2 h - a) uneven square I-beam | A = 1/2 (a + b) (d - h) + h t Mitsubishi symbol-like lamina | A = (sqrt(3) a^2)/2 Christmas tree with straight branches lamina | A = (6/11 + 316633/(23120 sqrt(3))) a^2

    30-60-90 triangle | x^_ = (a/6, a/(2 sqrt(3))) 3, 4, 5 triangle | x^_ = ((4 a)/3, a) diamond | x^_ = (0, 0) equilateral triangle | x^_ = (0, 0) inverted equilateral triangle | x^_ = (0, 0) golden rectangle | x^_ = (0, 0) golden rhombus | x^_ = (0, 0) golden triangle | x^_ = (0, 1/6 sqrt(5 + 2 sqrt(5)) a) half rectangle | x^_ = (0, b/4) half square | x^_ = (0, a/4) hexagram | x^_ = (0, 0) isosceles pentagon | x^_ = (0, (a d^2 + 2 h (d + h) w)/(3 a d + 6 h w)) isosceles right pentagon | x^_ = (0, 1/3 (h + b^2/(b + h))) isosceles right triangle | x^_ = (a/3, a/3) isosceles trapezoid | x^_ = (0, ((2 a + b) h)/(3 (a + b))) isosceles triangle | x^_ = (0, h/3) Kepler triangle | x^_ = ((sqrt(ϕ) a)/3, a/3) kite | x^_ = (1/3 (-sqrt(a^2 - h^2) + sqrt(b^2 - h^2)), 0) lozenge | x^_ = (1/4 (2 + sqrt(2)) a, a/(2 sqrt(2))) monokite | x^_ = (0, (7 a)/6) octagram | x^_ = (0, 0) parallelogram | x^_ = (1/2 (cos(A) a + b), 1/2 sin(A) a) pennant | x^_ = (h/3, 0) pentagram | x^_ = (0, 0) quarter rectangle | x^_ = (a/4, b/4) quarter square | x^_ = (a/4, a/4) rectangle | x^_ = (0, 0) regular decagon | x^_ = (0, 0) regular diamond | x^_ = (0, 0) regular heptagon | x^_ = (0, 0) regular hexagon | x^_ = (0, 0) regular nonagon | x^_ = (0, 0) regular octagon | x^_ = (0, 0) regular pentagon | x^_ = (0, 0) rhombus | x^_ = (0, 0) right trapezoid | x^_ = ((a (h_1 + 2 h_2))/(3 (h_1 + h_2)), 1/3 (h_2 + h_1^2/(h_1 + h_2))) 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)) square | x^_ = (0, 0) star of Lakshmi | x^_ = (0, 0) trapezoid | x^_ = (b/2 + ((2 a + b) (c^2 - d^2))/(6 (-a^2 + b^2)), ((2 a + b) sqrt((a - b + c - d) (a - b - c + d) (a - b + c + d) (-a + b + c + d)))/(6 (-a^2 + b^2))) 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)) hat polykite | x^_ = ((3 a)/2, 0) inverted isosceles trapezoid | x^_ = (0, ((2 a + b) sqrt(-(-a + b)^2 + 4 c^2))/(6 (a + b))) inverted T-beam | x^_ = (0, (b (d - h)^2 + (2 d - h) h t)/(2 (b (d - h) + h t))) rectangular cross beam | x^_ = (0, 0) rectangular L-beam | x^_ = ((a^2 + (b - t) t)/(2 (a + b - t)), (b^2 + (a - t) t)/(2 (a + b - t))) square channel | x^_ = ((b^2 (d - h) + h t^2)/(2 (b (d - h) + h t)), 0) square cross | x^_ = (0, 0) square I-beam | x^_ = (0, 0) square L-beam | x^_ = ((a^2 + a t - t^2)/(4 a - 2 t), (a^2 + a t - t^2)/(4 a - 2 t)) square T-beam | x^_ = (0, (b d^2 - b h^2 + h^2 t)/(2 b d - 2 b h + 2 h t)) swastika | x^_ = (0, 0) tapered I-beam | x^_ = (0, 0) tilted square cross | x^_ = (0, 0) uneven square I-beam | x^_ = (0, (a (d - h)^2 + b (d - h) (3 d + h) + 4 d h t)/(4 (a + b) (d - h) + 8 h t)) Mitsubishi symbol-like lamina | x^_ = (0, 0) Christmas tree with straight branches lamina | x^_ = (0, -((68978520 + 175160359 sqrt(3)) a)/(51 (416160 + 3482963 sqrt(3))))