Geometry A. Right Triangle Legs of a right triangle : a, b Hypotenuse : c Altitude : h Medians : m a, m b, m c Angles :, Radius of circumscribed circle : R Radius of inscribed circle : r Area : S 1. + = 90 Figure 1. 2. sin = = cos 3. cos = = sin 4. tan = = cot 5. cot = = tan
6. sec = = cosec 7. cosec = = sec 8. Pythagorean Theorem a 2 + b 2 = c 2 9. a 2 = fc, b 2 = gc, where f and c are projections of the legs a and b, respectively, onto the hypotenuse c. Figure 2. 10. h 2 = fg, where h is the altitude from the right angle. 11. = b 2 -, = a 2 -, where m a and m b are the medians to the legs a and b. Figure 3.
12. m c =, where m c is the median to the hypotenuse c. 13. R = = m c 14. r = = 15. ab = ch 16. s = = B Isosceles Triangle Base : a Legs : b Base angle : Vertex angle : Altitude to the base : h Perimeter : L Area : S
Figure 4. 17. = 90-18. h 2 = b 2 19. L = a +2b 20. S = = sin C. Equilateral Triangle Side of a equilateral triangle : a Altitude : h Radius of circumscribed circle : R Radius of inscribed circle : r Perimeter : L Area : S Figure 5. 21. h =
22. R = h = 23. r = h = = 24. L = 3a 25. S = = D. Scalene Triangle ( A triangle with no two sides equal) Sides of a triangle : a, b, c Semiperimeter : p Angle of a triangle :,, Altitudes to the sides a, b, c : h a, h b, h c Medians to the sides a, b, c : m a, m b, m c Bisectors of the angles,, : t a, t b, t c Radius of circumscribed circle : R Radius of inscribed circle : r Area : S
Figure 6. 26. = 180 27. a + b > c, b + c > a, a + c > b. 28. < c, < a, < b. 29. Midline q =, q a. 30. Law of Cosines a 2 = b 2 + c 2 2bc cos b 2 = a 2 + c 2 2ac cos c 2 = a 2 + b 2 2ab cos Figure 7. 31. Law of Sines = = = 2R, where R is the radius of the circumscribed circle. 32. R = = = = = = =
33. r 2 =, = = =. 34. sin =, cos =, tan =. 35. h a =, h b =, h c =. 36. h a = b sin = c sin, h b = a sin = c sin, h c = a sin = b sin. 37. = -, = -, = -.
Figure 8. 38. AM = m a, BM = m b, CM = m c (Fig. 8). 39. =, =, =, 40. S = = =, S = = =, S = S = pr S = (Heron s Formula), S = 2R 2 sin sin sin, S = p 2 tan tan tan. E. Square Side of a square : a Diagonal : d
Radius of circumscribed circle : R Perimeter : L Area : S 41. d = a Figure 9. 42. R = = 43. r = 44. L = 4a 45. S = a 2 F. Rectangle Sides of a rectangle : a, b Diagonal : d Radius of circumscribed circle : R Perimeter : L
Area : S Figure 10. 46. d = 47. R = 48. L = 2(a + b) 49. S = ab G. Parallelogram Sides of a parallelogram : a, b Diagonals : d 1, d 2 Consecutive angles : Angle between the diagonals : Altitude : h Perimeter : L Area : S
Figure 11. 50. = 180 51. + = 2(a 2 + b 2 ) 52. h = b sin = b sin 53. L = 2(a + b) 54. S = ah = ab sin, S = d 1 d 2 sin. H. Rhombus Side of a rhombus : a Diagonals : d 1, d 2 Consecutive angles : Altitude : H Radius of inscribed circle : r Perimeter : L Area : S
55. = 180 Figure 12. 56. + = 4a 2 57. h = a sin = 58. r = = = 59. L = 4a 60. S = ah = a 2 sin S = d 1 d 2. I. Trapezoid Bases of a trapezoid : a, b Midline : q Altitude : h Area : S
61. q = Figure 13. 62. S = h = qh J. Isosceles Trapezoid Bases of a trapezoid : a, b Leg : c Midline : q Altitude : h Diagonal : d Radius of circumscribed circle : R Area : S
Figure 14. 63. q = 64. d = 65. h = 66. R = 67. S = h = qh K. Isosceles Trapezoid with Inscribed Circle Bases of a trapezoid : a, b Leg : c Midline : q Altitude : h Diagonal : d
Radius of circumscribed circle : R Perimeter : L Area : S 68. a + b = 2c Figure 15. 69. q = = c 70. d 2 = h 2 + c 2 71. r = = 72. R = = = = = 73. L = 2(a + b) = 4c 74. S = h = =qh = ch
3.12 Trapezoid with Inscribed Circle Bases of a trapezoid : a, b Lateral sides : c, d Midline : q Altitude : h Diagonals : d 1, d 2 Angle between the diagonals : Radius of inscribed circle : r Radius of circumscribed circle : R Perimeter : L Area : S 75. a + b = c + d Figure 16. 76. q = = 77. L = 2( a + b ) = 2 ( c + d ) 78. S = h = h
S = d 1 d 2 sin. L. Kite Sides of a kite : a, b Diagonals : d 1, d 2 Angles :,, Perimeter : L Area : S Figure 17. 79. + + 2 = 360 80. L = 2( a + b ) 81. S =
M. Cyclic Quadrilateral Sides of a quadrilateral : a, b, c, d Diagonals : d 1, d 2 Angles between the diagonals : Internal angles : Radius of circumscribed circle : R Perimeter : L Semiperimeter : p Area : S 82. = = 180 Figure 18. 83. Ptolemy s Theorem ac + bd = d 1 d 2 84. L = a + b + c + d
85. R =, where p =. 86. S = d 1 d 2 sin, S =, where p =. N. Tangential Quadrilateral Sides of a quadrilateral : a, b, c, d Diagonals : d 1, d 2 Angles between the diagonals : Radius of inscribed circle : r Perimeter : L Semiperimeter : p Area : S
Figure 19. 87. a + c = b + d 88. L = a + b + c + d = 2(a + c) = 2(b + d) 89. r =, where p =. 90. S = pr = d 1 d 2 sin O. General Quadrilateral Sides of a quadrilateral : a, b, c, d Diagonals : d 1, d 2 Angle between the diagonals : Internal angles : Perimeter : L Area : S
Figure 20. 91. = 360 92. L = a + b + c + d 93. S = d 1 d 2 sin P. Regular Hexagon Side : a Internal angle : Slant height : m Radius of inscribed circle : r Radius of circumscribed circle : R Perimeter : L Semiperimeter : p Area : S
Figure 21. 94. = 120 95. r = m = 96. R = a 97. L = 6a 98. S = pr = where p = Q. Regular Polygon Side : a Number of sides : n Internal angle :
Slant height : m Radius of inscribed circle : r Radius of circumscribed circle : R Perimeter : L Semiperimeter : p Area : S 99. = 180 Figure 22. 100. R = 101. r = m = = = 102. L = na
103. S = sin S = pr = p, where p =. R. Circle Radius : R Diameter : d Chord : a Secant segment : e, f Tangent segment : g Central angle : Inscribed angle : Perimeter : L Area : S 104. a = 2R sin
Figure 23. 105. a 1 a 2 = b 1 b 2 Figure 24. 106. ee 1 = ff 1
Figure 25. 107. g 2 = ff 1 Figure 26.
108. = Figure 27. 109. L = 2 R = d 110. S = R 2 = = S. Sector of a circle Radius of a circle : R Arc length : s Central angle ( in radians ) : x Perimeter : L Area : S
Figure 28. 111. s = Rx 112. s = 113. L = s + 2R 114. S = = = T. Segment of a circle Radius of a circle : R Arc length : s Chord : a Central angle ( in radians ) : x Central angle ( in degrees ) : Height of the segment : h Perimeter : L Area : S
Figure 29. 115. a = 2 116. h = R -, h < R 117. L = s + a 118. S = [ sr a(r h)] = ( - sin ) = (x sin x) S ha. U. Cube Edge : a Diagonal : d Radius of inscribed sphere : r Radius of circumscribed sphere : r Surface area : S Volume : V
Figure 30. 119. d = a 120. r = 121. R = 122. S = 6a 2 123. V = a 3 V. Rectangular Parallelepiped Edges : a, b, c Diagonal : d Surface area : S Volume : V
Figure 31. 124. d = 125. S = 2(ab + ac + bc) 126. V = abc W. Prism Lateral edge : l Height : h Lateral area : S L Area of base : S B Total surface area : S Volume : V
Figure 32. 127. S = S L + 2S B. 128. Lateral Area of a Right Prism S L = (a 1 + a 2 + a 3 + + a n )l 129. Lateral Area of an Oblique Prism S L = pl, 130. V = S B h where p is the perimeter of the cross section. 131. Cavalieri s Principle Given two solids included between parallel planes. If every plane cross section parallel to the given planes has the same area in both solids, then the volumes of the solids are equal.
X. Regular Tetrahedron Triangle side length : a Height : h Area of base : S B Surface area : S Volume : V Figure 33. 132. h = a 133. S B = 134. S = a 2 135. V = S B h =. Y. Regular Pyramid Side of base : a
Lateral edge :b Height : h Slant height : m Number of sides : n Semiperimeter of base : p Radius of inscribed sphere of base : r Area of base : S B Lateral surface area : S L Total surface area : S Volume : V Figure 34. 136. m = 137. h = 138. S L = nam = na = PM
139. S B = pr 140. S = S B + S L 141. V = S B h = prh Z. Frustum of a Regular Pyramid Base and top side lengths : a 1, a 2, a 3,, a n and b 1, b 2, b 3,, b n Height : h Slant height : m Area of bases : S 1, S 2 Lateral surface area : S L Perimeter of bases : P 1, P 2 Scale factor : k Total surface area : S Volume : V
Figure 35. 142. = = = = = = k 143. = k 2 144. S L = 145. S = S L + S 1 + S 2 146. V = (S 1 + + S 2 ) 147. V = [ 1 + + ( ) 2 ] = [ 1 + k + k 2 ] AA. Rectangular Right Wedge Sides of base : a, b Top edge : c
Height : h Lateral surface area : S L Area of base : S B Total surface area : S Volume : V 148. S L = (a + c) + b Figure 36. 149. S B = ab 150. S = S B + S L 151. V = (2a + c) BB. Platonic Solids Edge : a Radius of inscribed circle : r Radius of circumscribed circle : R
Surface area : S Volume : V 152. Five Platonic Solids The platonic solids are convex polyhedral with equivalent faces composed of congruent convex regular polygons. Solid Number Number Number Section of Vertices Of Edges Of Faces Tetrahedron 4 6 4 3.25 Cube 8 12 6 3.22 Octahedron 6 12 8 3.27 Icosahedron 12 30 20 3.27 Dodecahedron 20 30 12 3.27 Octahedron Figure 37. 153. r =
154. R = 155. S = 2a 2 156. V = Icosahedron Figure 38. 157. r = 158. R = 159. S = 5a 2 160. V = Dodecahedron
Figure 39. 161. r = 162. R = 163. S = 3a 2 164. V = CC. Right Circular Cylinder Radius of base : R Diameter of base : d Height : H Lateral surface area : S L Area of base : S B
Total surface area : S Volume : V Figure 40. 165. S L = 2 RH 166. S = S L + 2S B = 2 R(H + R) = d(h + ) 167. V = S B H = R 2 H DD. Right Circular Cylinder with an Oblique Plane Face Radius of base : R The greatest height of a side : h 1 The shortest height of a side : h 2 Lateral surface area : S L Area of plane end faces : S B Total surface area : S
Volume : V Figure 41. 168. S L = R(h 1 + h 2 ) 169. S B = R 2 + R 170. S = S L + S B = R [ h 1 + h 2 + R + ] 171. V = (h 1 + h 2 ) EE. Right Circular Cone Radius of base : R Diameter of base : d Height : H
Slant height : m Lateral surface area : S L Lateral surface area : S L Area of base : S B Total surface area : S Volume : V 172. H = Figure 42. 173. S L = Rm = 174. S B = R 2 175. S = S L + S B = R(m + R) = d ( m + ) 176. V = S B H = R 2 H
FF. Frustum of a Right Circular Cone Radius of bases : R, r Height : H Slant height : m Scale factor : k Area of bases : S 1, S 2 Lateral surface area : S L Total surface area : S Volume : V 177. H = Figure 43. 178. = k 179. = = k 2 180. S L = m(r + r)
181. S = S 1 + S 2 + S L = [R 2 + r 2 + m(r + r)] 182. V = (S 1 + + S 2 ) 183. V = [ 1 + + ( ) 2 ] = [ 1 + k + k 2 ] GG. Sphere Radius : R Diameter : d Surface area : S Volume : V 184. S = 4 R 2 Figure 44. 185. V = R 3 H = d 3 = SR HH. Spherical Cap
Radius of sphere : R Radius of base : r Height : h Area of plane face : S B Area of spherical cap : S C Total surface area : S Volume : V Figure 45. 186. R = 187. S B = r 2 188. S C = (h 2 + 2r 2 ) = (2Rh + r 2 ) 189. V = h 2 (3R h) = h(3r 2 + h 2 ) II. Spherical Sector Radius of sphere : R
Radius of base of spherical cap : r Height : h Total surface area : S Volume : V 190. S = R (2h + r) Figure 46. 191. V = R 2 h Note : The given formulas are correct both for open and closed spherical sector. JJ. Spherical Segment Radius of sphere : R Radius of bases : r 1, r 2
Height : h Area of spherical surface : S S Area of plane end faces : S 1, S 2 Total surface area : S Volume : V 192. S S = 2 Rh Figure 47. 193. S = S S + S 1 + S 2 = (2Rh + + ) 194. V = h(3 + 3 + h 2 ) KK. Spherical Wedge Radius : R Dihedral angle in degrees : x Dihedral angle in radians :
Area of spherical lune : S L Total surface area : S Volume : V Figure 48. 195. S L = = 2R 2 x 196. S = R 2 + = R 2 + 2R 2 x 197. V = = R 3 x LL. Ellipsoid Semi-axes : a, b, c Volume : V
Figure 49. 198. V = abc Prolate Spheroid Semi-axes : a, b, b ( a > b ) Surface area : S Volume : V 199. S = 2 b (b + ), where e =. 200. V = b 2 a Oblate Spheroid Semi-axes : a, b, b ( a < b ) Surface area : S Volume : V
201. S = 2 b ( b + ), where e =. 202. V = b 2 a MM. Circular Torus Major radius : R Minor radius : r Surface area : S Volume : V 203. S = 4 Rr Figure 50. 204. V = 2 Rr 2