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Reference Guide

Geometry - Reference Guides

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Description
permacharts TM Geometry Geometry QUADRILATERALS CIRCLES & CONICS GENERAL QUADRILATERALS A d Note: For the following figures, C = circumference and D s = arc length Sum of the Interior Angles A + B + C + D = 360° a θ CIRCLES Diameter Perimeter a + b + c + d p c D = 2R B q s Area Circumference 1 1 segment pqsinθ =−θ+b2 2d2 2 a c tan b 2πR = πD h 2 4 a 1 2 Area − − = + − 4pq2 2 2b22d a c  C b ▯ 4   πRπD= 1 θ R 4 sector Theorem: Diagonals of a quadrilateral with consecutive sides a, b, c, 2nd 2 ar2 2 Sector & Segment of Circle perpendicular if and only if ac = b + d b  a  θ = 2c2s−1  = sin   RECTANGLES R  2R D b C sR= θ Interior Angles hR=b− A = B = C = D = 90°  θ   a = 2 sin = 2 tan  p 2 2 Perimeter a a   1 2 2 2(a + b) bR= cos   = R c4 2 2 Area (sector) = Rs =1 R2θ Area 2 2 ab A b B 1 2 Area (segment) = 2R ( )− sin TRAPEZOIDS Sum of the Interior Angles D a C PARABOLAS (SEGMENT) A + B + C + D = 360°, a||b Arc Length   Perimeter h 1 2 2 b26  a + + a  B 2 b a+ +6 8aln b  ab+h + + 1 1    siθ β sin  θ β Area Area A b B a 2 1 3 ab 2h()+b PARALLELOGRAMS 1 1 Interior Angles D b A 2 b 2b C C ELLIPSES A = C, B = D Diameter A + B = 180° q 2a = major axis 2b = minor axis Interior Lengths ha A a B sin a a a Area 2 2 p πab p a bb aA 2 cos 2 2 h q abb B 2 cos Perimeter b A b B ab−2 Perimeter k = a 2(a + b) π 2 2 2 Area P = 4 ∫ d sin θθ bh = absinA = absinB 0 = 2π 1ab2 2  (approximately) 2  1 GEOMETRY • A-588-6 w w w . p e r m a c h a r t s . c o m © 1997-2012 Mindsource Technologies Inc. permachartsM TRIANGLES • For the following triangles, s = semiperimeter2(a + b + c), RIGHT TRIANGLES B r = radius of inscribed circle, R = radius of circumscribed circle Interior Angles Area A + B = C = 90° 1 1 c GENERAL TRIANGLES ab = hc Sum of the Interior Angles Radii 2 2 Radii Perimeter h a A + B + C = 180° ab sinC C ab a + b + c r= = () tan r= A Height cs2 abc ab c R = = R c1 b C h = asinB = bsinA 2sinC K C 2 Perimeter EQUILATERAL TRIANGLES C a + b + c Interior Angles Perimeter b a A = B = C = 60° 3a Area 1 1 abc h CcKah = = sin Radii Area a a 2 2 4 R h ==rs −s() () ()−s c − r=a1 3 h a a 3 = 1 6 4 2 1 A c B R=a 3 A a B 3 PLANE AREA APPROXIMATION FORMULAS REGULAR POLYGONS • Divide the planar area ABCD into n parallel strips of equal • For the following polygons, n = number of sides, r = radius of thickness h inscribed circle, and R = radius of circumscribed circle • Let 0 ,1x 2 x , …n x denote the lengths of the vertical cords C B • h = B− Xn n a R 2n r h θ X2 Interior Angles Perimeter n −2 na = 2nRsin180° X1 θ = n 180°,>n 2  n  D X A Length Radii Area 0 180 ° 1 180° 1 180° a = 2 tan  r = cot  na cot   Trapezoidal Rule  n  2  n  4  n  1 1  2 180 ° Area+=+hx x0 1 2 x 1 n−n  = 2Rsin180 ° R = 1 csc180° = nr tan  2 2   n  2  n   n  1 2 360° Simpson’s Rule (n even) = 2nR sin  n  1 Area =++h(+0 1 2 3 42 x 22x 14 x n−nn x 3 n Polygon r R Area 2 Durand’s Rule 3 Triangle 0.2887a 0.5774a 0.4330a 4 Square 0.5000a 0.7071a 1.0000a 2 4 11 11 4  2 A+r+=+h+10 x02110 x2 1x x n−n− n10 10  5 Pentagon 0.6882a 0.8506a 1.7205a 6 Hexagon 0.8660a 1.0000a 2.5981a 2 7 Heptagon 1.0383a 1.1524a 3.6339a 2 Weddle’s Rule (n = 6) 2 3 8 Octagon 1.2071a 1.3066a 4.8284a +rea+=++ hx(0 12 34 56 x x x x 9 Nonagon 1.3737a 1.4619a 6.1818a 2 10 10 Decagon 1.5388a 1.6180a 7.6942a 2 2 GEOMETRY • A-588-6 w w w.permacharts.com © 1997-2012 Mindsource Technologies Inc. permachartsM PLANE SOLIDS PARALLELEPIPED RIGHT
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