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SAT Math: Geometry and Trigonometry

Area, volume, lines, angles, triangles, circles, and right-triangle trigonometry.

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  1. 1.A kite brace is a right triangle. One leg is 9 inches and the other leg is 12 inches. What is the hypotenuse?

  2. 2.A greenhouse floor is a rectangle 14 feet long and 9 feet wide. What is the area of the floor?

  3. 3.A boat sail is a triangle with base 16 meters and height 7 meters. What is the area of the sail?

  4. 4.A circular pond has a radius of 6 meters. What is the area of the pond's surface?

  5. 5.A grain silo is a cylinder with radius 3 feet and height 10 feet. What is the volume of the silo?

  6. 6.A cylindrical tin holds 36 cubic inches of wax. A cone-shaped candle uses the same radius and the same height. How much wax does the candle use?

  7. 7.Two angles sit side by side on a straight hinge. One measures 47 degrees. What does the other measure?

  8. 8.Two ribbons cross. One angle at the crossing measures 118 degrees. What is the angle vertically opposite that one?

  9. 9.A roof truss is a triangle with two angles of 38 degrees and 67 degrees. What is the third angle?

  10. 10.A zip-line cable is the hypotenuse of a right triangle. The horizontal leg is 5 meters and the vertical leg is 12 meters. How long is the cable?

  11. 11.A guy wire meets level ground at 30 degrees. The wire is 20 feet long, and the pole is vertical. How high above the ground does the wire meet the pole?

  12. 12.A brace meets a floor at 45 degrees and forms a right triangle. The horizontal leg is 8 centimeters. How long is the vertical leg?

  13. 13.In a right triangle, one acute angle measures 30 degrees. What is the sine of that angle?

  14. 14.A support cable rises at 60 degrees from a deck and forms a right triangle with the deck and a post. What is the cosine of the 60-degree angle?

  15. 15.A stage backdrop is enlarged so that every length is doubled. What happens to its area?

  16. 16.A clay model of a fountain is enlarged so it is three times as tall, three times as wide, and three times as deep. By what factor does the volume of clay change?

  17. 17.A desk blotter is a parallelogram with base 11 inches and height 6 inches. What is its area?

  18. 18.A flower bed is a trapezoid with parallel sides of 8 feet and 14 feet. The height is 5 feet. What is the area?

  19. 19.A bicycle wheel has a radius of 9 inches. How far does a point on the rim travel in one full turn?

  20. 20.A garden fence goes around a rectangle that is 15 yards long and 8 yards wide. How long is the fence?

  21. 21.A picture frame corner is a right angle split into two adjacent angles. One of them is 28 degrees. What is the other?

  22. 22.An isosceles banner has a vertex angle of 40 degrees. The two base angles are equal. What is each base angle?

  23. 23.At a gate, an exterior angle of a triangle has remote interior angles of 35 degrees and 48 degrees. What is the exterior angle?

  24. 24.Two garden plots are similar. A side of 6 feet on the smaller plot corresponds to a side of 15 feet on the larger plot. A different side of the smaller plot is 10 feet. What is the corresponding side of the larger plot?

  25. 25.Two similar stained-glass panels have corresponding lengths in the ratio 2 to 5. The smaller panel has area 8 square centimeters. What is the area of the larger panel?

  26. 26.Two roof trusses each have an 8-foot side and an 11-foot side with a 42-degree angle between those sides. Which congruence test matches the given information?

  27. 27.Two metal brackets have matching side lengths of 5 inches, 7 inches, and 9 inches. Which test shows the brackets are congruent?

  28. 28.Two window triangles each have a 30-degree angle and an 80-degree angle, with a 14-centimeter side lying between those angles. Which congruence test matches?

  29. 29.Two right triangles each have a hypotenuse of 17 meters and a leg of 8 meters. Which test proves the triangles are congruent?

  30. 30.One triangle has angles of 41 degrees and 63 degrees. A second triangle has angles of 41 degrees and 76 degrees. No side lengths are given. Which conclusion is justified?

  31. 31.A parking lot has sides of 7 yards, 24 yards, and 25 yards. Which statement about the angle between the 7-yard side and the 24-yard side is true?

  32. 32.On a trail map, a ranger station is at (1, 2) and a cabin is at (4, 6). Each grid unit is 1 kilometer. What is the straight-line distance between them?

  33. 33.A park path runs from a bench at (2, 5) to a fountain at (10, 1). Which point is exactly halfway between them?

  34. 34.A round pizza has radius 10 inches. One slice has a central angle of 72 degrees. What is the length of the crust on that slice?

  35. 35.A lawn sprinkler waters a 90-degree sector of a circle with radius 6 feet. What area does it water?

  36. 36.A camera on the rim of a circular plaza sees an inscribed angle that intercepts an arc of 110 degrees. What is the measure of the inscribed angle?

  37. 37.A Ferris wheel seat moves along an arc of 84 degrees, and the angle is measured at the center of the wheel. What is that central angle?

  38. 38.A radius of a wagon wheel meets the ground at the single point where the wheel touches. The ground is tangent to the wheel. What angle does the radius make with the ground?

  39. 39.A triangle is drawn in a circle so that one side is a diameter and the opposite vertex lies on the circle. What is the angle at that vertex?

  40. 40.A triangular ornament has all three sides the same length. What is the measure of each interior angle?

  41. 41.Oak Street is parallel to Pine Street, and a trail crosses both. The angle above and to the right of the Oak Street crossing is 73 degrees. What is the corresponding angle above and to the right at Pine Street?

  42. 42.A ladder crosses two parallel balcony rails. The interior angle on the left of the ladder at the upper rail is 128 degrees. What is the alternate interior angle at the lower rail?

  43. 43.A brace crosses two parallel pipes. The interior angles on the same side of the brace are supplementary. One of them is 58 degrees. What is the other?

  44. 44.A diagonal slit crosses the two parallel long edges of a picture mat. The exterior angle north of the upper edge is 41 degrees. What is the alternate exterior angle south of the lower edge?

  45. 45.Two sides of a triangle are sticks of 6 inches and 9 inches. Which length can be the third side?

  46. 46.A shipping carton is a rectangular prism 7 inches by 6 inches by 4 inches. What is its volume?

  47. 47.All six faces of a cooler that is 6 inches by 4 inches by 3 inches are covered with vinyl. How much vinyl is used?

  48. 48.A tent is a pyramid with base area 30 square feet and height 9 feet. What is the volume inside the tent?

  49. 49.A globe has radius 6 centimeters. What is the volume of the globe?

  50. 50.A paper popcorn cone has radius 4 centimeters and height 9 centimeters. What is the volume of the cone?

  51. 51.A cylindrical can and a cone-shaped funnel both have radius 5 inches and height 12 inches. The can holds 300π cubic inches. What is the volume of the funnel?

  52. 52.A circular hatch cover has a diameter of 14 centimeters. What is the area of the cover?

  53. 53.A rectangular deck is 12 feet by 7 feet. A right triangular sandbox with legs 6 feet and 4 feet is cut out of one corner. What deck area remains?

  54. 54.A tailor cuts a wool strip that is 4 yards long. How many inches long is the strip?

  55. 55.A jump rope is 4 yards long. How many feet long is the rope?

  56. 56.Each side of a square quilt block is 13 inches. What is the area of the block?

  57. 57.A square floor tile has area 81 square centimeters. What is the length of one side?

  58. 58.A square napkin is cut along a diagonal, making a right triangle with two legs of 7 centimeters each. How long is the diagonal?

  59. 59.A square patio has a diagonal of 10√2 meters. What is the side length of the patio?

  60. 60.A roof triangle is a 30-60-90 triangle. The side opposite the 30-degree angle is 5 inches. What is the side opposite the 60-degree angle?

  61. 61.Another roof triangle is a 30-60-90 triangle. The side opposite 60 degrees is 6√3 centimeters. What is the hypotenuse?

  62. 62.A steep trail forms a right triangle with a horizontal run and a vertical drop of 7 meters. The sloping path is 25 meters. What is the sine of the angle between the path and the horizontal run?

  63. 63.A wheelchair ramp is a right triangle. The ramp surface is 17 feet, and the horizontal leg adjacent to the angle with the ground is 8 feet. What is the cosine of that angle?

  64. 64.A ladder, the ground, and a wall form a right triangle with legs 9 feet and 40 feet. What is the tangent of the angle opposite the 9-foot leg?

  65. 65.A right triangle has a 30-degree angle and a hypotenuse of 14 centimeters. How long is the side opposite the 30-degree angle?

  66. 66.A right triangle has a 60-degree angle and a hypotenuse of 18 meters. How long is the side adjacent to the 60-degree angle?

  67. 67.A shelf bracket is a right triangle with a 45-degree angle. The side opposite that angle is 11 inches. How long is the adjacent side?

  68. 68.A loading ramp rises at 60 degrees from the floor and forms a right triangle. What is the sine of the 60-degree angle?

  69. 69.A sunshade makes a 30-degree angle with a horizontal bar and closes a right triangle. What is the cosine of that 30-degree angle?

  70. 70.A folded greeting card stands as a right triangle with a 30-degree angle at the table. What is the tangent of that 30-degree angle?

  71. 71.An attic roof rises at 60 degrees from the floor in a right triangle. What is the tangent of the 60-degree angle?

  72. 72.A square board is sawed along a diagonal, and one acute angle of a resulting right triangle is 45 degrees. What is the cosine of that angle?

  73. 73.One corner of a kite is a right triangle with a 45-degree angle. What is the sine of that 45-degree angle?

  74. 74.A television screen is a rectangle with a 25-inch diagonal and a width of 15 inches. What is the height of the screen?

  75. 75.A parade banner is a right triangle with legs of 18 feet and 24 feet. What is the hypotenuse?

  76. 76.A scarf is a right triangle with perpendicular sides of 10 inches and 14 inches. What is the area of the scarf?

  77. 77.A circular hoop has circumference 16π inches. What is the radius of the hoop?

  78. 78.A round rug has area 100π square feet. What is the radius of the rug?

  79. 79.A round plate has a radius of 11 centimeters. What is the diameter of the plate?

  80. 80.A toy chest and a full-size chest are similar. Every edge of the full-size chest is twice the matching edge of the toy chest. The full-size chest holds how many times as much as the toy chest?

  81. 81.A town map is reprinted so that every distance on the new map is three times the matching distance on the old map. A park's area on the new map is how many times its area on the old map?

  82. 82.A small right triangle has sides 5 feet, 12 feet, and 13 feet. A similar triangle has shortest side 15 feet. What is the longest side of the larger triangle?

  83. 83.Two cottage roofs are similar triangles. One angle on the smaller roof is 52 degrees. What is the corresponding angle on the larger roof?

  84. 84.A triangular herb garden has a perimeter of 14 meters. A similar garden is built with every length three times as long. What is the perimeter of the new garden?

  85. 85.A rectangular door is 8 feet wide and 15 feet tall. What is the distance between opposite corners of the door?

  86. 86.A window pane is a rhombus with sides of 10 centimeters and a height of 6 centimeters. What is the area of the pane?

  87. 87.A triangular flag has area 48 square inches and a height of 8 inches. What is the base?

  88. 88.A round pipe has an outer diameter of 9 inches. What is the circumference of the outer circle?

  89. 89.An inscribed angle in a circular window measures 37 degrees. What is the measure of its intercepted arc?

  90. 90.A triangular brace has a right angle, and one of the acute angles is 33 degrees. What is the other acute angle?

  91. 91.Four cables meet at a ring and fill the space around the point. Three of the angles between neighboring cables are 80 degrees, 95 degrees, and 70 degrees. What is the fourth angle?

  92. 92.Two parallel handrails are crossed by one post. The same-side interior angles are in the ratio 1 to 4. What is the larger angle?

  93. 93.An equilateral triangle frame has a perimeter of 36 centimeters. What is the length of one side?

  94. 94.An isosceles steeple triangle has two equal sides of 13 feet and a base of 10 feet. What is the height from the top vertex to the base?

  95. 95.An isosceles flower triangle has equal sides of 10 meters and a base of 12 meters. What is its area?

  96. 96.A water trough is a triangular prism. The triangular end has area 20 square inches, and the trough is 9 inches long. What is the volume?

  97. 97.A cylindrical water tank has volume 75π cubic feet and radius 5 feet. What is the height of the tank?

  98. 98.A cone-shaped party hat has volume 20π cubic inches and radius 2 inches. What is the height of the hat?

  99. 99.Two similar rugs have corresponding lengths in the ratio 1 to 3. The smaller rug has area 7 square feet. What is the area of the larger rug?

  100. 100.Two similar bottles have corresponding lengths in the ratio 1 to 2. The smaller bottle holds 5 cups. How much does the larger bottle hold?

  101. 101.From a point outside a circular fountain, two tangent segments touch the circle. One tangent segment is 9 centimeters. How long is the other tangent segment?

  102. 102.A circular table has radius 10 inches. A chord of the table is 16 inches long. How far is the chord from the center of the table?

  103. 103.Two inscribed angles in the same circle intercept the same arc of 100 degrees. What is the measure of each inscribed angle?

  104. 104.In a triangular roof truss, a brace joins the midpoints of two sides. The third side is 22 feet long. How long is the brace?

  105. 105.A four-sided tabletop has three angles of 80 degrees, 95 degrees, and 110 degrees. What is the fourth angle?

  106. 106.A floor tile is a regular pentagon, so all five sides and all five interior angles are equal. What is each interior angle?

  107. 107.A regular hex nut has six congruent exterior angles, one at each vertex, and those exterior angles go all the way around the nut. What is one exterior angle?

  108. 108.A rectangular meadow is 20 yards wide and 21 yards long. What is the straight-line distance between opposite corners?