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T E C H N I C A L D ATA S H E E T formulae (equilateral triangle, pentagon, hexagon, octagon, polygon) equilateral triangle 2 a 3 A = 4 a a h h = 3 2 a 5 A = r2 10 + 2 5 pentagon 8 C 1 a = r 10 - 2 5 2 1 B E ! = r 6 + 2 5 4 A ! D Construction: a r AB = 0.5 r, BC = BD , CD = CE 3 hexagon A = a2 3 2 d = 2 a s 2 s s = = d 1.155 a 3 3 s = d = 0.866 d 2 3 A = a s = 0.83 s2 octogan 2 = 2 s d 2 = s2 a = s x tan 22.5° ~ ~ 0.451 s d a s s = d x cos 22.5° ~ ~ 0.942 d s d = cos 22.5° ~ ~ 1.083 s polygon A = A1 + A2 + A3 b A3 h1 3 h ah1 + bh2 + bh3 A1 A2 = 2 2 h Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (circle, annulus, sector of a circle, segment of a circle, ellipse) " d2 " r2 circle A = 4 = ~ ~ 0.785 d2 " r " d r d U = 2 = annulus " (D2 = d 2) A = 4 = " (d + b) b b d D - d b = 2 D " r2 r a r2 sector of a circle A = 360° = 2 br = 2 ! r " ra b = 180° " a = 180° a (a = a in circular measure) b a segment of a circle s = 2 r s in 2 2 h r ! r A = 6s ( 3 h2 + 4 s2 ) = 2 ( a - sina) h 2 s r = + 3 2 8h a s a h = r ( 1 - cos ) = 2 tan 4 h 2 " a = 180° a ( a = a in circular measure ) b " A = Dd = " a b ellipse 2 a D + d " 2 U ~ ~ 2 d 0.785 b 1 1 1 = " (a + b) 1 + #2 + #4 + 256 #6 + d 4 64 25 a-b 16384 #8 + … , where # = a+b D Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (square, rectangle, parallelogram, trapezium, triangle) square A = a2 c = A a d d = a 2 rectangle A = a b d b d = a2 + b2 a parallelogram A = a b = a b sin! d1 ( a + h cot !) 2 + h 2 d1 = h d2 ! ( a - h cot !) 2 + h 2 d2 = a trapezium a + b a A = h = m h 2 a + b h m = m 2 b ah triangle A = = " s 2 = s ( s - a) (s - b) (s - c) c b h a+b+c " s = 2 a Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (cylinder, hollow cylinder, cone, frustum of cone, sphere) ! cylinder V = d2 h 4 r Am = ! r h h 2 A0 = 2 ! r (r + h) d hollow cylinder ! V = h (D 2 - d 2 ) h 4 d D ! cone V = r2h 3 A2 Vm = ! r m x h A0 = ! r (r + m) m m = h2 + r2 A2 : = A1 = x 2 h 2 r A1 ! V = h ( D 2 + Dd + d 2 ) frustum of cone 12 m ! m(D+d)=2!ph m 2 Am = h 2 d D-d p m = ( 2 )2 + h 2 D 4 1 V = ! r2 = !d2 sphere 3 6 ~ 4 . 189 r3 d ~ r A0 = 4 ! r3 = ! d 2 Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (cube, cuboid, parallelepiped, pyramid, frustum of pyramid) cube V = a2 h 2 A0 = 6a a d d = 3 a a a cuboid V = a b c A0 = (a b + a c + b c) a d d = a2 + b2 + c2 a a parallelepiped V = A1 h h (Cavalieri principle) A1 pyramid h A1 h V = 3 A1 frustum of pyramid A2 h 2 2 V = ( A +A + A1 A2 ) 3 h ~ ~ h A1 + A2 2 A1 Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (zone of sphere, segment of sphere, sector of sphere, sphere with cylindrical bore, sphere with conical bore) zone of a sphere ! a V = 6 h ( 3a 2+ 3b 2+ h 2) h Am = 2 ! r h A0 = ! ( 2 r h + a2 + b 2 ) b r ! 3 segment of a sphere V = 6 h ( 4 s2 + h2 ) = ! h (r- h ) h 3 Am = 2 ! r h ! r = ( s2 + 4h 2 ) s 4 sector of a sphere 2 V = ! r h 2 r 3 ! r(4h+ s) A0 = h 2 s sphere with cylindrical boring ! V = h3 6 h A0 = 2 ! h (R+r) R r sphere with conical boring 2 V = ! r h 3 3 h 2 A0 = 2 ! h (h + r2 = h r 4 Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400 T E C H N I C A L D ATA S H E E T formulae (torus, sliced cylinder, ungula, barrel, prismoid) torus ! 2 V = 4 Dd2 d A0 = !2 D d D sliced cylinder ! V = 4 d2h d d ungula 2 V = 3 r2h Am = 2 r h d A0 = A 0 + ! r 2 + ! r r2 + h 2 2 2 d barrel ! V = 12 h ( 2 D 2+ d 2) h D d prismoid h V = ( A1 +A2+ 4A) 6 A2 This formula may be used for calculations A h involving solids shown on formulae cube and thus speres and parts of spheres. h 2 A1 Revision: 01-05-08 a.b.e. is an ISO 9001 : 2000 Registered Company. a.b.e. Construction Chemicals Limited P.O. Box 23053, Isipingo, 4110 7 Wilcox Road, Isipingo, 4133 Tel. (031) 913 5400