Geometry. A. Right Triangle. Legs of a right triangle : a, b. Hypotenuse : c. Altitude : h. Medians : m a, m b, m c. Angles :,

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1 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 sin = = cos 3. cos = = sin 4. tan = = cot 5. cot = = tan

2 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 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.

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

4 Figure = h 2 = b 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 h =

5 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

6 Figure = 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 Law of Sines = = = 2R, where R is the radius of the circumscribed circle. 32. R = = = = = = =

7 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. = -, = -, = -.

8 Figure 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

9 Radius of circumscribed circle : R Perimeter : L Area : S 41. d = a Figure 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

10 Area : S Figure 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

11 Figure = = 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

12 55. = 180 Figure = 4a 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

13 61. q = Figure 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

14 Figure 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

15 Radius of circumscribed circle : R Perimeter : L Area : S 68. a + b = 2c Figure q = = c 70. d 2 = h 2 + c r = = 72. R = = = = = 73. L = 2(a + b) = 4c 74. S = h = =qh = ch

16 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 q = = 77. L = 2( a + b ) = 2 ( c + d ) 78. S = h = h

17 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 = L = 2( a + b ) 81. S =

18 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 Ptolemy s Theorem ac + bd = d 1 d L = a + b + c + d

19 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

20 Figure 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

21 Figure = 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

22 Figure = r = m = 96. R = a 97. L = 6a 98. S = pr = where p = Q. Regular Polygon Side : a Number of sides : n Internal angle :

23 Slant height : m Radius of inscribed circle : r Radius of circumscribed circle : R Perimeter : L Semiperimeter : p Area : S 99. = 180 Figure R = 101. r = m = = = 102. L = na

24 103. 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

25 Figure a 1 a 2 = b 1 b 2 Figure ee 1 = ff 1

26 Figure g 2 = ff 1 Figure 26.

27 108. = Figure 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

28 Figure 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

29 Figure a = 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

30 Figure d = a 120. r = 121. R = 122. S = 6a V = a 3 V. Rectangular Parallelepiped Edges : a, b, c Diagonal : d Surface area : S Volume : V

31 Figure 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

32 Figure S = S L + 2S B Lateral Area of a Right Prism S L = (a 1 + a 2 + a 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 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.

33 X. Regular Tetrahedron Triangle side length : a Height : h Area of base : S B Surface area : S Volume : V Figure h = a 133. S B = 134. S = a V = S B h =. Y. Regular Pyramid Side of base : a

34 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 m = 137. h = 138. S L = nam = na = PM

35 139. 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

36 Figure = = = = = = k 143. = k S L = 145. S = S L + S 1 + S V = (S S 2 ) 147. V = [ ( ) 2 ] = [ 1 + k + k 2 ] AA. Rectangular Right Wedge Sides of base : a, b Top edge : c

37 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 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

38 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 Cube Octahedron Icosahedron Dodecahedron Octahedron Figure r =

39 154. R = 155. S = 2a V = Icosahedron Figure r = 158. R = 159. S = 5a V = Dodecahedron

40 Figure r = 162. R = 163. S = 3a 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

41 Total surface area : S Volume : V Figure 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

42 Volume : V Figure 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

43 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 S L = Rm = 174. S B = R S = S L + S B = R(m + R) = d ( m + ) 176. V = S B H = R 2 H

44 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 = k 179. = = k S L = m(r + r)

45 181. S = S 1 + S 2 + S L = [R 2 + r 2 + m(r + r)] 182. V = (S S 2 ) 183. V = [ ( ) 2 ] = [ 1 + k + k 2 ] GG. Sphere Radius : R Diameter : d Surface area : S Volume : V 184. S = 4 R 2 Figure V = R 3 H = d 3 = SR HH. Spherical Cap

46 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 R = 187. S B = r 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

47 Radius of base of spherical cap : r Height : h Total surface area : S Volume : V 190. S = R (2h + r) Figure 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

48 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 S = S S + S 1 + S 2 = (2Rh + + ) 194. V = h( h 2 ) KK. Spherical Wedge Radius : R Dihedral angle in degrees : x Dihedral angle in radians :

49 Area of spherical lune : S L Total surface area : S Volume : V Figure 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

50 Figure V = abc Prolate Spheroid Semi-axes : a, b, b ( a > b ) Surface area : S Volume : V 199. S = 2 b (b + ), where e = V = b 2 a Oblate Spheroid Semi-axes : a, b, b ( a < b ) Surface area : S Volume : V

51 201. S = 2 b ( b + ), where e = V = b 2 a MM. Circular Torus Major radius : R Minor radius : r Surface area : S Volume : V 203. S = 4 Rr Figure V = 2 Rr 2

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