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GRE Quantitative Reasoning: Geometry

Angles, triangles, circles, area, and volume, with figures described in words.

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  1. 1.In triangle PQR, angle P measures 47 degrees and angle Q measures 68 degrees. What is the degree measure of angle R?

  2. 2.Angles JKL and LKM are adjacent and together form straight angle JKM. Angle JKL measures 112 degrees. What is the degree measure of angle LKM?

  3. 3.Parallel lines m and n are cut by transversal t. One angle formed by t and m corresponds to an angle formed by t and n. The angle on m measures 73 degrees. What is the degree measure of the corresponding angle on n?

  4. 4.Two lines intersect and form vertical angles. One of those angles measures 54 degrees. Quantity A is the degree measure of the vertical angle opposite that angle. Quantity B is 54.

  5. 5.Triangle ABC is isosceles with AB = AC. The vertex angle at A measures 48 degrees. What is the degree measure of each base angle?

  6. 6.Triangle DEF is equilateral. What is the degree measure of each interior angle?

  7. 7.Right triangle GHI has a right angle at H. Leg GH is 9 and leg HI is 12. What is the length of hypotenuse GI?

  8. 8.In a 30-60-90 triangle, the side opposite the 30 degree angle has length 7. Quantity A is the length of the side opposite the 60 degree angle. Quantity B is 7√3.

  9. 9.In a 30-60-90 triangle, the side opposite the 30 degree angle has length 8. What is the length of the hypotenuse?

  10. 10.An isosceles right triangle has a right angle and two legs of length 5. What is the length of the hypotenuse?

  11. 11.Two triangles are similar. The smaller triangle has sides 3, 4, and 5. The side of length 3 corresponds to a side of length 9 on the larger triangle. What is the length of the larger side that corresponds to the side of length 4?

  12. 12.A circle has radius 1. Quantity A is the numerical value of its area. Quantity B is the numerical value of its circumference.

  13. 13.Triangle JKL is congruent to triangle MNO, with J corresponding to M, K to N, and L to O. Angle K measures 41 degrees. What is the degree measure of angle N?

  14. 14.Quadrilateral WXYZ has interior angles of 80 degrees, 95 degrees, 110 degrees, and x degrees. What is x?

  15. 15.Diagonals drawn from one vertex divide a convex pentagon into 3 triangles. What is the sum of the pentagon's interior angles, in degrees?

  16. 16.In triangle ABC, AB = 6 and AC = 9. The length of BC is not given, and no angle is identified as a right angle. Quantity A is the length of BC. Quantity B is 10.

  17. 17.A convex hexagon has interior angles of 90, 100, 110, 120, 130, and x degrees. What is x?

  18. 18.What is the degree measure of each interior angle of a regular octagon?

  19. 19.A rectangular poster is 14 centimeters long and 6 centimeters wide. What is its area in square centimeters?

  20. 20.Parallel lines p and q are cut by transversal r. One interior angle between the parallel lines measures 61 degrees. Quantity A is the degree measure of the alternate interior angle, on the opposite side of r and also between the parallel lines. Quantity B is 61.

  21. 21.A parallelogram has base 11 and a height of 7 drawn perpendicular to that base. What is the area of the parallelogram?

  22. 22.A triangle has base 10 and a height of 9 drawn perpendicular to that base. What is the area of the triangle?

  23. 23.A trapezoid has parallel bases of lengths 8 and 14 and a height of 5 between those bases. What is the area of the trapezoid?

  24. 24.In triangle ABC, angle A measures 40 degrees. No other angle or side is given. Quantity A is the degree measure of angle B. Quantity B is the degree measure of angle C.

  25. 25.A circle has radius 6. What is its circumference?

  26. 26.A circle has radius 5. What is its area?

  27. 27.A circle has radius 9. A central angle of 40 degrees cuts an arc of the circle. What is the length of that arc?

  28. 28.A cube has edges of length 3. Quantity A is the length of a diagonal of one square face. Quantity B is the length of a segment that joins one vertex to the opposite vertex through the interior of the cube.

  29. 29.A circle has radius 6. What is the area of a sector whose central angle is 60 degrees?

  30. 30.A polygon has area 8. A second polygon is a scaled copy in which every length is 3 times as long. What is the area of the second polygon?

  31. 31.A solid has volume 5. A similar solid is built so that every length is twice as long. What is the volume of the larger solid?

  32. 32.A right circular cylinder has radius 3 and height 4. Quantity A is the volume of the cylinder. Quantity B is 36π.

  33. 33.A rectangular prism is 7 units long, 5 units wide, and 2 units tall. What is its volume?

  34. 34.A closed rectangular box has edge lengths 6, 3, and 2. What is its surface area?

  35. 35.A cube has edges of length 4. What is its volume?

  36. 36.A cube has edges of length 5. Quantity A is the numerical value of the volume. Quantity B is the numerical value of the surface area.

  37. 37.A right circular cylinder has radius 3 and height 10. What is its volume?

  38. 38.A right circular cylinder has radius 4 and height 7. The lateral surface is the curved part and does not include the two bases. What is that lateral surface area?

  39. 39.A right circular cylinder has radius 3 and height 6. What is its total surface area, including the curved side and both circular bases?

  40. 40.A 30-60-90 triangle has hypotenuse 18. Quantity A is the length of the side opposite the 30 degree angle. Quantity B is 9.

  41. 41.A 30-60-90 triangle has hypotenuse 14. What is the length of the side opposite the 60 degree angle?

  42. 42.A 45-45-90 triangle has hypotenuse 8√2. What is the length of each leg?

  43. 43.A right triangle has legs of lengths 8 and 15. What is the length of the hypotenuse?

  44. 44.Lines u and v are cut by transversal w. The lines are not stated to be parallel. One angle formed by w and u measures 70 degrees. Quantity A is the degree measure of the angle at v in the corresponding position. Quantity B is 70.

  45. 45.In triangle ABC, side BC is extended beyond C to point D, so angle ACD is an exterior angle. The remote interior angles at A and B measure 28 degrees and 47 degrees. What is the degree measure of angle ACD?

  46. 46.Two lines intersect. One of the four angles measures 123 degrees. What is the degree measure of an angle adjacent to it?

  47. 47.Parallel lines are cut by a transversal. One interior angle between the parallel lines measures 118 degrees. What is the degree measure of the same-side interior angle, the other interior angle on the same side of the transversal?

  48. 48.Triangle ABC has AB = 10 and AC = 10. Side BC is not given. Quantity A is the degree measure of angle BAC, the angle between the equal sides. Quantity B is 60.

  49. 49.Two triangles are similar. The larger has sides 15, 20, and 25. The side of length 15 corresponds to a side of length 6 on the smaller triangle. What is the length of the smaller side that corresponds to the side of length 20?

  50. 50.Triangle ABC is congruent to triangle DEF, with A corresponding to D, B to E, and C to F. AB = 7, BC = 9, and CA = 10. What is the perimeter of triangle DEF?

  51. 51.A convex heptagon has interior angles of 100, 110, 120, 130, 140, 175, and x degrees. What is x?

  52. 52.A rectangle has perimeter 28. Its length and width are not given separately. Quantity A is the area of the rectangle. Quantity B is 48.

  53. 53.Each exterior angle of a regular pentagon measures 72 degrees. What is the degree measure of each interior angle?

  54. 54.A triangle has a base of 16 and a height of 7 drawn perpendicular to that base. What is the area of the triangle?

  55. 55.A circle has diameter 12. What is its area?

  56. 56.A sector of a circle has arc length 4π. The radius and the central angle are not given separately. Quantity A is the radius of the circle. Quantity B is 8.

  57. 57.A circle has diameter 10. A central angle of 72 degrees cuts an arc. What is the length of that arc?

  58. 58.A circle has diameter 8. What is the area of a sector whose central angle is 45 degrees?

  59. 59.Two garden plots are similar. A side of the smaller plot and the corresponding side of the larger plot are in the ratio 2 to 5. The smaller plot has area 12. What is the area of the larger plot?

  60. 60.A 45-45-90 triangle has legs of length 6. Quantity A is the length of one leg. Quantity B is the length of the hypotenuse.

  61. 61.In parallelogram ABCD, angle A measures 68 degrees. Angle B is consecutive to angle A. What is the degree measure of angle B?

  62. 62.A rectangle is 5 units wide and 12 units long. Each corner is a right angle. What is the length of a diagonal?

  63. 63.An equilateral triangle has sides of length 6. What is the length of an altitude from one vertex to the opposite side?

  64. 64.Triangle ABC has AB = 8 and BC = 8. Side AC is not given. Quantity A is the length of AC. Quantity B is 8.

  65. 65.A circle has circumference 20π. What is its radius?

  66. 66.A circle has area 49π. What is its radius?

  67. 67.A ring is the region between two circles with the same center. The outer radius is 5 and the inner radius is 3. What is the area of the ring?

  68. 68.Two triangles are similar. A side of the smaller triangle has length 4 and corresponds to a side of length 10 on the larger triangle. The smaller triangle has area 8. Quantity A is the area of the larger triangle. Quantity B is 80.

  69. 69.Two rectangular prisms are similar. Each edge of the larger prism is twice the corresponding edge of the smaller prism. The smaller prism has volume 9. What is the volume of the larger prism?

  70. 70.A storage cube has edges of length 6. What is its surface area?

  71. 71.A rectangular prism has edges 3, 4, and 12, and those edges meet at right angles. What is the length of a space diagonal from one vertex through the interior to the opposite vertex?

  72. 72.A right circular cylinder has volume 48π. The radius and the height are not given separately. Quantity A is the radius. Quantity B is 4.

  73. 73.The angles of a triangle measure x degrees, 2x degrees, and 3x degrees. What is the degree measure of the smallest angle?

  74. 74.Triangle LMN is isosceles with LM = LN. Base angle LMN measures 55 degrees. What is the degree measure of vertex angle MLN?

  75. 75.Parallel lines are cut by a transversal. One exterior angle, outside the parallel lines, measures 41 degrees. What is the degree measure of the same-side exterior angle, on the same side of the transversal and also outside the parallel lines?

  76. 76.A square and an equilateral triangle each have perimeter 24. Quantity A is the side length of the square. Quantity B is the side length of the equilateral triangle.

  77. 77.A square has a diagonal of length 10. Each corner of the square is a right angle. What is the area of the square?

  78. 78.A right triangle has legs 7 and 24. What is its perimeter?

  79. 79.A square patio of side 10 has a square garden of side 4 removed from its interior. The garden lies entirely inside the patio. What is the remaining area of the patio?

  80. 80.A triangle has area 24. The base and the corresponding height are not given separately. Quantity A is that height. Quantity B is 6.

  81. 81.A circle has radius 10. An arc of the circle has length 5π. What is the degree measure of the central angle that cuts that arc?

  82. 82.An inscribed angle has its vertex on a circle, and its sides pass through the endpoints of an arc that measures 80 degrees. What is the degree measure of the inscribed angle?

  83. 83.A central angle has its vertex at the center of a circle and intercepts an arc of 110 degrees. What is the degree measure of the central angle?

  84. 84.A circle has radius 10. A central angle of 36 degrees intercepts an arc. Quantity A is the length of that arc. Quantity B is 4π.

  85. 85.The interior angles of a regular nonagon, a polygon with 9 sides, sum to 1260 degrees. What is the degree measure of each interior angle?

  86. 86.A convex hexagon has one exterior angle at each vertex, formed by extending one side. What is the sum of those six exterior angles, in degrees?

  87. 87.Each exterior angle of a regular polygon measures 24 degrees. How many sides does the polygon have?

  88. 88.A regular polygon has an exterior angle of 40 degrees at each vertex. Quantity A is the degree measure of each interior angle. Quantity B is 140.

  89. 89.A semicircular region has radius 8. What is its area?

  90. 90.A semicircular region has radius 5. Its boundary includes the curved edge and the straight diameter. What is the perimeter of that boundary?

  91. 91.A quarter circle has radius 12. Its central angle is the right angle of 90 degrees. What is the area of the quarter circle?

  92. 92.A rectangle has area 36. Its length and width are not given separately. Quantity A is the perimeter of the rectangle. Quantity B is 30.

  93. 93.A cube has volume 125. What is the length of one edge?

  94. 94.A right circular cylinder has volume 100π and radius 5. What is its height?

  95. 95.A closed cylindrical can is made of metal. The can has radius 2 and height 5, and the metal covers the curved wall, the top, and the bottom. What is the total area of that metal?

  96. 96.A circular rug has radius 6. Quantity A is the number of square units in the area of the rug. Quantity B is the number of units in the circumference of the rug.

  97. 97.Right triangle ABC has its right angle at C. Angle A measures 37 degrees. What is the degree measure of angle B?

  98. 98.Angles 1 and 2 are adjacent and together form a right angle. Angle 1 measures 28 degrees. What is the degree measure of angle 2?

  99. 99.A triangle has two sides of lengths 6 and 11. Which of the following could be the length of the third side?

  100. 100.In triangle ABC, AB = 7, BC = 9, and AC = 12. Which angle is the largest?

  101. 101.Triangle ABC has base BC of length 20 and altitude 12 from A to BC. A line parallel to BC intersects AB and AC and cuts off a smaller triangle that shares angle A. That smaller triangle is similar to triangle ABC, and its altitude from A is 3. What is the length of the smaller base?

  102. 102.Two similar triangles have corresponding sides in the ratio 1 to 4, with 1 on the smaller triangle. The larger triangle has area 48. What is the area of the smaller triangle?

  103. 103.Two similar solids have corresponding lengths in the ratio 1 to 3, with 1 on the smaller solid. The larger solid has volume 54. What is the volume of the smaller solid?

  104. 104.Two polygons are similar. Each side of the larger polygon is 3 times the corresponding side of the smaller polygon. The smaller polygon has perimeter 14. What is the perimeter of the larger polygon?

  105. 105.Triangle XYZ is an isosceles right triangle. The right angle is at Y, and XY = YZ. What is the degree measure of angle X?

  106. 106.In a 30-60-90 triangle, the side opposite the 60 degree angle has length 6√3. What is the length of the hypotenuse?

  107. 107.A rhombus has diagonals of lengths 6 and 8. The diagonals are perpendicular and bisect each other. What is the area of the rhombus?

  108. 108.A triangle has sides 13, 14, and 15. What is its area?