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    Concave Laminae

    Named laminae

    annulus | arbelos | disk with square hole | disk with equilateral triangular hole | elliptical annulus | equilateral triangle with circular hole | hexagram | octagram | pentagram | rectangle with rectangular hole | shield | square with circular hole | square with square hole | star of Lakshmi | hat polykite | inverted T-beam | lune | rectangular cross beam | rectangular L-beam | salinon | ... (total: 65)

    Definitions

    Definitions Annulus

    Definitions Arbelos

    Definitions Disk with square hole

    Defining inequalities

    b^2<=x^2 + y^2<=a^2

    x^2 + y^2<=a^2/4 and y>=0 and y^2>=(a/2 + x) (1/2 a (2 r - 1) - x) and 1/4 a^2 (2 r - 1) + x^2 + y^2>=a r x

    x^2 + y^2<=a^2 and abs(x)>=b/2 or abs(y)>=b/2

    sqrt(3) b<=3 (sqrt(3) x + y) or sqrt(3) (b + 3 x)<=3 y or sqrt(3) b + 6 y<=0 and x^2 + y^2<=a^2

    r^2<=x^2/a^2 + y^2/b^2<=1

    sqrt(3) (a + 3 x)>=3 y and sqrt(3) a + 6 y>=0 and sqrt(3) a>=3 (sqrt(3) x + y) and x^2 + y^2>=b^2

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

    -sqrt(5 + 7/sqrt(2))<=(5 sqrt(2) x + 7 x + 2 sqrt(2) y + 3 y)/a<=sqrt(5 + 7/sqrt(2)) and -sqrt(5 + 7/sqrt(2))<=(2 sqrt(2) x + 3 x + 5 sqrt(2) y + 7 y)/a<=sqrt(5 + 7/sqrt(2)) and -sqrt(5 + 7/sqrt(2))<=(2 sqrt(2) x + 3 x - 5 sqrt(2) y - 7 y)/a<=sqrt(5 + 7/sqrt(2)) and -sqrt(5 + 7/sqrt(2))<=(5 sqrt(2) x + 7 x - 2 sqrt(2) y - 3 y)/a<=sqrt(5 + 7/sqrt(2)) or (24 + 17 sqrt(2)) x>=sqrt(58 + 41 sqrt(2)) a + (10 + 7 sqrt(2)) y and sqrt(116 + 82 sqrt(2)) a + (34 + 24 sqrt(2)) y>=2 (7 + 5 sqrt(2)) x and 7 sqrt(2) x + 10 x + 17 sqrt(2) y + 24 y<=sqrt(58 + 41 sqrt(2)) a or 17 sqrt(2) x + 24 x + 7 sqrt(2) y + 10 y>=sqrt(58 + 41 sqrt(2)) a and sqrt(116 + 82 sqrt(2)) a + 2 (7 + 5 sqrt(2)) y>=(34 + 24 sqrt(2)) x and sqrt(58 + 41 sqrt(2)) a + (10 + 7 sqrt(2)) x>=(24 + 17 sqrt(2)) y or 7 sqrt(2) x + 10 x + 17 sqrt(2) y + 24 y>=sqrt(58 + 41 sqrt(2)) a and 24 sqrt(2) x + 34 x + 10 sqrt(2) y + 14 y<=sqrt(116 + 82 sqrt(2)) a and sqrt(58 + 41 sqrt(2)) a + (24 + 17 sqrt(2)) x>=(10 + 7 sqrt(2)) y or (24 + 17 sqrt(2)) y>=sqrt(58 + 41 sqrt(2)) a + (10 + 7 sqrt(2)) x and 10 sqrt(2) x + 14 x + 24 sqrt(2) y + 34 y<=sqrt(116 + 82 sqrt(2)) a and sqrt(58 + 41 sqrt(2)) a + 17 sqrt(2) x + 24 x + 7 sqrt(2) y + 10 y>=0 or (10 + 7 sqrt(2)) y>=sqrt(58 + 41 sqrt(2)) a + (24 + 17 sqrt(2)) x and sqrt(116 + 82 sqrt(2)) a + 2 (7 + 5 sqrt(2)) x>=(34 + 24 sqrt(2)) y and sqrt(58 + 41 sqrt(2)) a + 7 sqrt(2) x + 10 x + 17 sqrt(2) y + 24 y>=0 or sqrt(58 + 41 sqrt(2)) a + 17 sqrt(2) x + 24 x + 7 sqrt(2) y + 10 y<=0 and sqrt(116 + 82 sqrt(2)) a + (34 + 24 sqrt(2)) x>=2 (7 + 5 sqrt(2)) y and sqrt(58 + 41 sqrt(2)) a + (24 + 17 sqrt(2)) y>=(10 + 7 sqrt(2)) x or sqrt(58 + 41 sqrt(2)) a + 7 sqrt(2) x + 10 x + 17 sqrt(2) y + 24 y<=0 and sqrt(116 + 82 sqrt(2)) a + 24 sqrt(2) x + 34 x + 10 sqrt(2) y + 14 y>=0 and sqrt(58 + 41 sqrt(2)) a + (10 + 7 sqrt(2)) y>=(24 + 17 sqrt(2)) x or sqrt(58 + 41 sqrt(2)) a + (24 + 17 sqrt(2)) y<=(10 + 7 sqrt(2)) x and sqrt(116 + 82 sqrt(2)) a + 10 sqrt(2) x + 14 x + 24 sqrt(2) y + 34 y>=0 and 17 sqrt(2) x + 24 x + 7 sqrt(2) y + 10 y<=sqrt(58 + 41 sqrt(2)) a

    2 a + 3 sqrt(5) x + 5 x>=sqrt(2 (5 + sqrt(5))) y and 2 a + sqrt(50 + 22 sqrt(5)) y>=(5 + sqrt(5)) x and sqrt(5) a + 2 sqrt(5) x + 2 sqrt(5 + 2 sqrt(5)) y<=a or sqrt(5) a + 2 sqrt(5) x + 2 sqrt(5 + 2 sqrt(5)) y>=a and a + 2 sqrt(5) x<=sqrt(5) a + 2 sqrt(5 + 2 sqrt(5)) y and 2 sqrt(5 + 2 sqrt(5)) y<=a and 3 sqrt(5) x + 5 x + sqrt(2 (5 + sqrt(5))) y<=2 a or a + 2 sqrt(5) x>=sqrt(5) a + 2 sqrt(5 + 2 sqrt(5)) y and 2 a + (5 + sqrt(5)) x + sqrt(50 + 22 sqrt(5)) y>=0 and 3 sqrt(5) x + 5 x + sqrt(2 (5 + sqrt(5))) y<=2 a or sqrt(2/(65 + 29 sqrt(5))) a + sqrt(5) y>=3 y and 3 sqrt(5) x + 5 x + sqrt(2 (5 + sqrt(5))) y>=2 a and sqrt(50 - 22 sqrt(5)) x + y<=2 sqrt(2/(65 + 29 sqrt(5))) a + sqrt(5) y or 2 sqrt(2/(65 + 29 sqrt(5))) a + sqrt(50 - 22 sqrt(5)) x + sqrt(5) y>=y and 2 a + 3 sqrt(5) x + 5 x<=sqrt(2 (5 + sqrt(5))) y or 2 sqrt(5 + 2 sqrt(5)) y>=a and sqrt(650 - 290 sqrt(5)) a + 10 sqrt(5 - 2 sqrt(5)) x + 20 y>=10 sqrt(5) y and 10 (sqrt(5 - 2 sqrt(5)) x + (sqrt(5) - 2) y)<=sqrt(650 - 290 sqrt(5)) a

    abs(x)<=a/2 and abs(y)<=b/2 and abs(x)>=c/2 or abs(y)>=d/2

    Alternate names

    annular disk | disk with circular hole | hollow circle

    shoemaker's knife

    Magen David | Mogen David | shield of David | six-point star | six-pointed star | Solomon's seal | star of David

    eight-point star

    five-point star | pentacle | pentalpha | pentangle

    hollow rectangle

    inverted square T-beam

    crescent

    Greek cross

    fylfot | gammadion

    Lamina properties

    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))) 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) 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))) rectangle with rectangular hole | (-a/2, -b/2) | (a/2, -b/2) | (a/2, b/2) | (-a/2, b/2) | (-c/2, -d/2) | (c/2, -d/2) | (c/2, d/2) | (-c/2, d/2) shield | (0, -1/2 sqrt(-D^2 + 4 R^2)) | (0, h - 1/2 sqrt(-d^2 + 4 r^2)) | ((4 (d - D) h^2 + (d + D) (d^2 - D^2 - 4 r^2 + 4 R^2) + 2 h sqrt((d^2 + 2 d D + D^2 + 4 h^2 - 4 r^2 + 8 r R - 4 R^2) (-d^2 - 2 d D - D^2 - 4 h^2 + 4 r^2 + 8 r R + 4 R^2)))/(4 ((d + D)^2 + 4 h^2)), -((d + D) sqrt((d^2 + 2 d D + D^2 + 4 h^2 - 4 r^2 + 8 r R - 4 R^2) (-d^2 - 2 d D - D^2 - 4 h^2 + 4 r^2 + 8 r R + 4 R^2)) - 2 h ((d + D)^2 + 4 (h^2 - r^2 + R^2)))/(4 ((d + D)^2 + 4 h^2))) | (-(4 (d - D) h^2 + (d + D) (d^2 - D^2 - 4 r^2 + 4 R^2) + 2 h sqrt((d^2 + 2 d D + D^2 + 4 h^2 - 4 r^2 + 8 r R - 4 R^2) (-d^2 - 2 d D - D^2 - 4 h^2 + 4 r^2 + 8 r R + 4 R^2)))/(4 ((d + D)^2 + 4 h^2)), -((d + D) sqrt((d^2 + 2 d D + D^2 + 4 h^2 - 4 r^2 + 8 r R - 4 R^2) (-d^2 - 2 d D - D^2 - 4 h^2 + 4 r^2 + 8 r R + 4 R^2)) - 2 h ((d + D)^2 + 4 (h^2 - r^2 + R^2)))/(4 ((d + D)^2 + 4 h^2))) square with square hole | (-a/2, -a/2) | (a/2, -a/2) | (a/2, a/2) | (-a/2, a/2) | (-b/2, -b/2) | (b/2, -b/2) | (b/2, b/2) | (-b/2, b/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) 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 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)

    hexagram | 12 octagram | 16 pentagram | 10 rectangle with rectangular hole | 8 shield | 4 square with square hole | 8 star of Lakshmi | 16 hat polykite | 13 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

    annulus | 0<b<a arbelos | a>0 and 0<r<1 disk with square hole | 0<b<sqrt(2) a disk with equilateral triangular hole | 0<b<sqrt(3) a elliptical annulus | a>0 and b>0 and 0<r<1 equilateral triangle with circular hole | 0<b<2 sqrt(3) a hexagram | a>0 octagram | a>0 pentagram | a>0 rectangle with rectangular hole | 0<c<a and 0<d<b square with circular hole | 0<2 b<a square with square hole | 0<b<a star of Lakshmi | a>0 hat polykite | a>0 inverted T-beam | 0<t<b and 0<h<d lune | R>0 and r>0 and (d + r - R) (d - r + R) (-d + r + R)>0 rectangular cross beam | 0<t<b and 0<s<d rectangular L-beam | 0<t<a and 0<t<b salinon | 0<r<R square channel | 0<t<b and 0<h<d square cross | 0<a<h square I-beam | 0<t<b and 0<h<d square L-beam | 0<t<a square T-beam | 0<h<d and 0<t<b swastika | 0<3 t<a tapered I-beam | 0<t<b and 0<l<h<d tilted square cross | 0<a<h triquetra | a>0 uneven square I-beam | 0<t<a and 0<t<b and 0<h<d Acura symbol-like lamina | a>0 Android symbol-like lamina | a>0 Apple symbol-like lamina | a>0 Apple with Steve Jobs bite symbol-like lamina | a>0 batman insignia-like lamina | a>0 Bizarro insignia-like lamina | a>0 Chrysler symbol-like lamina | a>0 Honda symbol-like lamina | a>0 Kawasaki symbol-like lamina | a>0 McDonald's symbol-like lamina | a>0 Mercedes-Benz symbol-like lamina | a>0 Mitsubishi symbol-like lamina | a>0 Playboy bunny symbol-like lamina | a>0 Shell Oil symbol-like lamina | a>0 Subaru symbol-like lamina | a>0 Superman insignia-like lamina | a>0 Toyota symbol-like lamina | a>0 Twitter bird symbol-like lamina | a>0 Volkswagen symbol-like lamina | a>0 Yamaha symbol-like lamina | a>0 American flag lamina | a>0 bunny lamina | a>0 Christmas tree with curly branches lamina | a>0 Christmas tree with straight branches lamina | a>0 first Easter egg | a>0 second Easter egg | a>0 third Easter egg | a>0 fourth Easter egg | a>0 Temple menorah lamina | a>0 Temple menorah with candles lamina | a>0 Hanukkah menorah lamina | a>0 shamrock lamina | a>0

    Mitsubishi symbol-like lamina | (3 a)/2

    annulus | A = π (a^2 - b^2) arbelos | A = 1/4 π a^2 (1 - r) r disk with square hole | A = π a^2 - b^2 disk with equilateral triangular hole | A = π a^2 - (sqrt(3) b^2)/4 elliptical annulus | A = π a b (1 - r^2) equilateral triangle with circular hole | A = (sqrt(3) a^2)/4 - π b^2 hexagram | A = a^2/sqrt(3) octagram | A = (6 sqrt(2) - 8) a^2 pentagram | A = 1/4 sqrt(650 - 290 sqrt(5)) a^2 rectangle with rectangular hole | A = a b - c d square with circular hole | A = a^2 - π b^2 square with square hole | A = a^2 - b^2 star of Lakshmi | A = 2 (2 - sqrt(2)) a^2 hat polykite | A = 8 sqrt(3) a^2 inverted T-beam | A = b (d - h) + h t lune | A = r^2 cos^(-1)((-d^2 - r^2 + R^2)/(2 d r)) - R^2 cos^(-1)((d^2 - r^2 + R^2)/(2 d R)) + 1/2 sqrt((d + r - R) (d - r + R) (-d + r + R) (d + r + R)) rectangular cross beam | A = b s + t (d - s) rectangular L-beam | A = t (a + b - t) salinon | A = 1/4 π (r + R)^2 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) triquetra | A = 1/2 (2 π - sqrt(3)) a^2 uneven square I-beam | A = 1/2 (a + b) (d - h) + h t batman insignia-like lamina | A = a^2 (955/48 - 2/7 (-3 sqrt(10) + 2 sqrt(33) + 7 π + 3 sqrt(10) π) + 21 cos^(-1)(3/7) + 21 cos^(-1)(4/7)) Kawasaki symbol-like lamina | A = a^2 (271817/1427150 + (1691 sqrt(7/10510))/16854 - (33 sqrt(3310123/44713))/40000 - (14 sqrt(48525107/44713))/65025 + (61 sqrt(22817731/1051))/3468000 + (2831 sqrt(186707671/1051))/67416000 - sqrt(1159/5282731) π - sqrt(1159/5282731) cos^(-1)((3 sqrt(174217/61))/200) + 3/40 sqrt(89/1051) cos^(-1)((2 sqrt(6230))/159) - sqrt(1159/5282731) sec^(-1)(255 sqrt(61/2553953)) + 3/40 sqrt(89/1051) sec^(-1)(510/sqrt(256379)) + 3/20 sqrt(89/1051) sec^(-1)(3180/sqrt(2097839))) Mercedes-Benz symbol-like lamina | A = 1/16 (3 sqrt(3) + 2 π) a^2 Mitsubishi symbol-like lamina | A = (sqrt(3) a^2)/2 Toyota symbol-like lamina | A = a^2 (-(231 sqrt(22737/2))/341125 + (33 sqrt(377 (8415 sqrt(55145) - 70493)))/2776625 + (6 sqrt(377/5 (98385584627 sqrt(55145) - 20930698974873)))/2467086845 - (51 sqrt(377 (4275492467895 sqrt(55145) - 441413049303013)))/123354342250 + (66 sqrt(39 (407 sqrt(224545) - 23285)))/2184625 - (37 sqrt(39 (153884554703 sqrt(224545) - 65983338793565)))/38180691125 - (4 sqrt(4919300620377 sqrt(224545) - 418670161729755))/7636138225 - (29 sqrt(667/10 (493 sqrt(358609) - 82769)))/281025 + (17 sqrt(667/10 (99836227573 sqrt(358609) - 53703419997209)))/2106001350 + sqrt(667/10 (109629455653 sqrt(358609) - 64984651941449))/210600135 - (29 sqrt(138/5 (1073 sqrt(17462545) - 1054697)))/3845875 + (3 sqrt(138 (5187195699061 sqrt(17462545) - 6010365584154637)))/118326036125 - 282828/125 sqrt(4807/(34272102535994485 + 8202017542045 sqrt(17462545))) + (29 sqrt(897/10 (5213773 + 1537 sqrt(33013465))))/7783375 - (3 sqrt(6279/2 (12095230099350743 + 3899412753779 sqrt(33013465))))/484647411125 - 2031861/125 sqrt(966/(44065649502140335 + 7678672117205 sqrt(33013465))) + ((5355 - 200 sqrt(26) - 510 sqrt(58) + 106 sqrt(130)) π)/10000 + 51/250 sqrt(29/2) tan^(-1)(26/(53 sqrt(33))) - 53/250 sqrt(13/10) tan^(-1)((2 sqrt(1885/33))/51) + 1/25 sqrt(13/2) tan^(-1)((28 sqrt(1749))/2465) + 53/500 sqrt(13/10) tan^(-1)((28 sqrt(8745))/769) + 1/25 sqrt(13/2) tan^(-1)(2/669 sqrt(58/5 (39183 + 187 sqrt(55145)))) - 51/500 sqrt(29/2) tan^(-1)((2 sqrt(26 (751523 + 8415 sqrt(55145))))/20557) - (37 tan^(-1)((2 sqrt(390 (163915 + 407 sqrt(224545))))/9277))/(250 sqrt(10)) - 1/25 sqrt(13/2) tan^(-1)((4 sqrt(2442 sqrt(224545) - 70590))/17573) + 51/500 sqrt(29/2) tan^(-1)((2 sqrt(115 (1886279 + 4437 sqrt(358609))))/24743) + 3/50 sqrt(23/2) tan^(-1)((2 sqrt(145 (2572679 + 4437 sqrt(358609))))/8783) - (37 tan^(-1)(2/637 sqrt(69 (4482967 + 1073 sqrt(17462545)))))/(250 sqrt(10)) + 3/50 sqrt(23/2) tan^(-1)((4 sqrt(3 (5365 sqrt(17462545) - 4310413)))/155323) - 53/500 sqrt(13/10) tan^(-1)(2/983 sqrt(69 (8830477 + 1537 sqrt(33013465)))) - 3/50 sqrt(23/2) tan^(-1)((2 sqrt(39 (28113089 + 7685 sqrt(33013465))))/297451)) Yamaha symbol-like lamina | A = a^2 (271817/1427150 + (1691 sqrt(7/10510))/16854 - (33 sqrt(3310123/44713))/40000 - (14 sqrt(48525107/44713))/65025 + (61 sqrt(22817731/1051))/3468000 + (2831 sqrt(186707671/1051))/67416000 - sqrt(1159/5282731) π - sqrt(1159/5282731) cos^(-1)((3 sqrt(174217/61))/200) + 3/40 sqrt(89/1051) cos^(-1)((2 sqrt(6230))/159) - sqrt(1159/5282731) sec^(-1)(255 sqrt(61/2553953)) + 3/40 sqrt(89/1051) sec^(-1)(510/sqrt(256379)) + 3/20 sqrt(89/1051) sec^(-1)(3180/sqrt(2097839))) Christmas tree with straight branches lamina | A = (6/11 + 316633/(23120 sqrt(3))) a^2 jack-o'-lantern face lamina | A = (16 π - 103/16) a^2 Temple menorah lamina | A = (a^2 (3985 - 108 sqrt(7) + 48 sqrt(11) - 64 sqrt(33) - 84 sqrt(43) + 32 sqrt(110) - 16 sqrt(143) + 36 sqrt(195) + 12 sqrt(391) + 16 sqrt(1023) - 16 sqrt(1155) + 12 sqrt(4087) - 12 sqrt(4615) - 8464 sin^(-1)(1/23) + 7056 sin^(-1)(1/21) - 16384 sin^(-1)(31/32) + 8464 sin^(-1)((7 sqrt(43))/46) - 7056 sin^(-1)(sqrt(195)/14) + 18496 sin^(-1)(sqrt(4087)/68) - 16384 sin^(-1)((3 (sqrt(28609) - 31))/2048) - 2304 tan^(-1)(sqrt(391/185)) + 2304 tan^(-1)(3 sqrt(7)) + 1600 csc^(-1)(10) - 2304 csc^(-1)(12) + 16384 csc^(-1)(32) - 18496 csc^(-1)(34) - 1600 csc^(-1)(20/sqrt(391)) + 2304 csc^(-1)(24/sqrt(391)) - 18496 csc^(-1)(68/sqrt(4087)) + 18496 csc^(-1)(68/sqrt(4615))))/17424 smiley face lamina | A = π (a^2 + 2 r^2 - 2 R^2) + r^2 (-sec^(-1)(-(2 r (d_l - d_u))/((d_l - d_u)^2 + r^2 - R^2))) + R^2 sec^(-1)((2 R (d_l - d_u))/((d_l - d_u)^2 - r^2 + R^2)) - 1/2 sqrt((d_l - d_u + r - R) (d_l - d_u - r + R) (d_l - d_u + r + R) (-d_l + d_u + r + R))

    annulus | x^_ = (0, 0) arbelos | x^_ = (1/2 a (-1/2 + r), a/π) disk with square hole | x^_ = (0, 0) disk with equilateral triangular hole | x^_ = (0, 0) elliptical annulus | x^_ = (0, 0) equilateral triangle with circular hole | x^_ = (0, 0) hexagram | x^_ = (0, 0) octagram | x^_ = (0, 0) pentagram | x^_ = (0, 0) rectangle with rectangular hole | x^_ = (0, 0) square with circular hole | x^_ = (0, 0) square with square hole | x^_ = (0, 0) star of Lakshmi | x^_ = (0, 0) hat polykite | x^_ = ((3 a)/2, 0) 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))) salinon | x^_ = (0, -(2 (r - R))/π) 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) triquetra | 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)) batman insignia-like lamina | x^_ = (0, (3 (-244469 - 44800 sqrt(10) + 16128 sqrt(33) + 10 (6277 + 1344 sqrt(10) - 441 sqrt(33)) π) a)/(140 (-6685 - 288 sqrt(10) + 192 sqrt(33) + 96 (7 + 3 sqrt(10)) π - 7056 cos^(-1)(3/7) - 7056 cos^(-1)(4/7)))) Chrysler symbol-like lamina | x^_ = (0, 0) Mercedes-Benz symbol-like lamina | x^_ = (0, 0) 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)))) jack-o'-lantern face lamina | x^_ = (0, -(367 a)/(824 - 2048 π))

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