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ACT Math: Geometry

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

ACT is a registered trademark of ACT, Inc. FreeExam is not affiliated with ACT, Inc., and these questions are original practice, not an official test.

  1. 1.A cabinet door swings away from a right-angled frame. The door makes a 28 degree angle with one side of that frame. How many degrees separate the door from the other side of the right angle?

  2. 2.A straight railing is crossed by a brace, so two adjacent angles form a straight line. One of those angles measures 115 degrees. What is the other angle?

  3. 3.Two metal rods cross and form four angles. One angle measures 73 degrees. What is the measure of the angle vertically opposite it?

  4. 4.In triangle LMN, angle L measures 58 degrees and angle M measures 47 degrees. What is the measure of angle N?

  5. 5.In triangle ABC, angle A is 42 degrees and angle B is 67 degrees. Side BC is extended past C, forming an exterior angle at C. What is that exterior angle?

  6. 6.Parallel lines m and n are cut by a transversal. One angle formed by line m and the transversal measures 124 degrees. What is the corresponding angle formed by line n and the transversal?

  7. 7.Parallel lines j and k are cut by a transversal. One interior angle between the parallel lines measures 68 degrees. What is the alternate interior angle on the other side of the transversal?

  8. 8.Parallel shelves are cut by a diagonal brace. One same-side interior angle between the shelves measures 53 degrees. What is the other same-side interior angle?

  9. 9.Triangle DEF is isosceles with DE equal to DF. Angle E measures 72 degrees. What is the measure of angle D?

  10. 10.A yield sign is an equilateral triangle, so its three sides are equal and its three angles are equal. What is the measure of each angle?

  11. 11.Quadrilateral WXYZ has angle measures of 80 degrees, 95 degrees, and 110 degrees at W, X, and Y. What is the measure of angle Z?

  12. 12.Angle PQR measures 148 degrees. Ray QS bisects angle PQR, splitting it into two equal angles. What is the measure of angle PQS?

  13. 13.Right triangle RST has its right angle at S. Angle R measures 37 degrees. What is the measure of angle T?

  14. 14.Three adjacent angles sit on a straight line and fill that straight angle. Two of them measure 40 degrees and 65 degrees. What is the third angle?

  15. 15.A floor tile is a regular hexagon, with six equal sides and six equal interior angles. What is the measure of each interior angle?

  16. 16.A badge is a regular pentagon, with five equal sides and five equal interior angles. What is the measure of each interior angle?

  17. 17.Two strings cross. One angle at the crossing measures 41 degrees. What is the measure of an angle adjacent to it?

  18. 18.Three adjacent angles around a point fill a full circle. Two of them measure 95 degrees and 140 degrees. What is the third angle?

  19. 19.A ladder leans against a vertical wall and forms a right triangle with the wall and the ground. The foot of the ladder is 16 feet from the wall, and the top meets the wall 30 feet above the ground. How long is the ladder?

  20. 20.A loading ramp is the hypotenuse of a right triangle. The ramp rises 5 meters while the horizontal leg along the ground is 12 meters. How long is the ramp?

  21. 21.A park route goes 6 blocks east and then 8 blocks north. A straight walk joins the start to the finish and is the hypotenuse of that right triangle. How long is the straight walk?

  22. 22.A kite string is pulled straight. The kite is 24 feet above the reel, and the reel is 7 feet from the point on the ground directly under the kite. Those lengths are the legs of a right triangle. How long is the string?

  23. 23.A boat sail is a right triangle. One leg is 8 feet and the other leg is 15 feet. What is the length of the hypotenuse?

  24. 24.A right triangular flag has legs of 21 inches and 28 inches. What is the hypotenuse of the flag?

  25. 25.A diagonal beam crosses a rectangular opening, forming a right triangle with legs of 20 inches and 21 inches. How long is the beam?

  26. 26.A guy wire runs straight from the top of a 40-foot pole to a stake 9 feet from the base of the pole. The pole and the ground meet at a right angle. How long is the wire?

  27. 27.A hiking trail is the hypotenuse of a right triangle. The horizontal leg is 12 meters and the vertical leg is 16 meters. How long is the trail?

  28. 28.A right triangle marked on a field has legs of 12 yards and 35 yards. What is the hypotenuse?

  29. 29.A right triangle has a hypotenuse of 26 meters and one leg of 10 meters. What is the length of the other leg?

  30. 30.A right triangle has sides of 15 yards, 36 yards, and 39 yards. What is the perimeter of the triangle?

  31. 31.A triangular garden has a base of 14 feet and a height of 9 feet measured perpendicular to that base. What is the area of the garden?

  32. 32.A strip of bunting is cut as a triangle with a base of 10 inches and a height of 6 inches. What is the area of the bunting?

  33. 33.An isosceles cloth triangle has two equal sides of 10 cm and a base of 12 cm. The height to the base meets the base at its midpoint, so each half of the base is 6 cm. What is the area of the cloth?

  34. 34.Triangle ABC is similar to triangle DEF. Side AB corresponds to side DE. AB is 6 cm and DE is 15 cm. Side BC corresponds to side EF, and BC is 8 cm. What is the length of EF?

  35. 35.In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. Segment MN joins those midpoints and is parallel to side BC. BC is 18 cm. What is the length of MN?

  36. 36.An isosceles flower bed has two equal sides of 13 meters and a base of 10 meters. What is the perimeter of the bed?

  37. 37.A right triangular sail has legs of 18 feet and 24 feet. The hypotenuse is 30 feet. What is the area of the sail?

  38. 38.Isosceles triangle JKL has JK equal to JL. The vertex angle at J measures 40 degrees. What is the measure of base angle K?

  39. 39.A 30-60-90 triangle has its shortest side, the side opposite the 30 degree angle, equal to 7 inches. What is the hypotenuse?

  40. 40.A 30-60-90 triangle is cut from card stock. The hypotenuse is 18 cm. What is the length of the side opposite the 30 degree angle?

  41. 41.A circular fountain has a radius of 7 meters. What is the circumference of the fountain?

  42. 42.A round table has a radius of 8 feet. What is the area of the tabletop?

  43. 43.A bicycle wheel is a circle with a diameter of 10 inches. What is the circumference of the wheel?

  44. 44.A circular rug has a diameter of 14 feet. What is the area of the rug?

  45. 45.A circular running track has a circumference of 28 pi meters. What is the radius of the track?

  46. 46.A circular patio has an area of 81 pi square feet. What is the radius of the patio?

  47. 47.A circle has a radius of 12 cm. A central angle of 90 degrees cuts off one arc of the circle. What is the length of that arc?

  48. 48.A circle has a radius of 6 inches. A central angle of 120 degrees marks a sector of the circle. What is the area of the sector?

  49. 49.A circle has an inscribed angle that opens onto an arc of 80 degrees. The inscribed angle is formed by two chords meeting on the circle. What is the measure of that inscribed angle?

  50. 50.A central angle of a circle is formed by two radii and intercepts an arc of 110 degrees. What is the measure of the central angle?

  51. 51.Triangle UVW is inscribed in a circle. Side UW is a diameter of the circle, and vertex V lies on the circle. What is the measure of the angle at V?

  52. 52.A circular walkway surrounds a circular lawn. The lawn has radius 3 meters, and the outer edge of the walkway is a larger circle with radius 5 meters. What is the area of the walkway alone?

  53. 53.A circle has a radius of 13 inches. A chord of the circle is 5 inches from the center, and the segment from the center to the chord is perpendicular to the chord. How long is the chord?

  54. 54.A circular pond has a radius of 9 yards. What is the circumference of the pond?

  55. 55.A pie slice is a sector of a circle. The sector has a 120 degree central angle and a radius of 3 inches. What is the area of the slice?

  56. 56.A circular banner has a radius of 15 cm. A central angle of 120 degrees marks an arc along the rim. What is the length of that arc?

  57. 57.A circular stage has a radius of 13 feet. What is the diameter of the stage?

  58. 58.A circular hoop has a diameter of 22 inches. What is the circumference of the hoop?

  59. 59.A classroom floor is a rectangle 15 feet long and 8 feet wide. What is the area of the floor?

  60. 60.A window frame is a rectangle 13 inches long and 7 inches wide. What is the perimeter of the frame?

  61. 61.A floor tile is a square with a side length of 11 cm. What is the area of the tile?

  62. 62.A mat is a square with a side length of 15 cm. What is the perimeter of the mat?

  63. 63.A shop sign is a parallelogram with a base of 13 cm and a height of 6 cm perpendicular to that base. What is the area of the sign?

  64. 64.A patio is a trapezoid. The two parallel sides are 8 feet and 14 feet, and the height between those sides is 5 feet. What is the area of the patio?

  65. 65.An L-shaped countertop is made by cutting a 4 foot by 2 foot rectangular notch from one corner of a 10 foot by 6 foot rectangle. What is the area of the countertop?

  66. 66.A small rectangular poster is 4 inches wide and 6 inches tall. A similar poster is enlarged by a scale factor of 3. What is the area of the enlarged poster?

  67. 67.A logo is a triangle with area 20 square centimeters. A similar logo is printed so that every length is twice as long. What is the area of the printed logo?

  68. 68.A rectangular banner is 4 yards long. How many inches long is the banner?

  69. 69.One side of a rectangular picture border is 5 feet long. How many inches long is that side?

  70. 70.A triangular pennant has side lengths of 7 inches, 9 inches, and 11 inches. What is the perimeter of the pennant?

  71. 71.A rectangular pool cover has an area of 96 square meters and a length of 12 meters. What is the width of the cover?

  72. 72.A square patio has a perimeter of 36 feet. What is the length of one side?

  73. 73.A courtyard is a square measuring 10 meters on each side. A circular fountain with radius 3 meters sits entirely inside the square. What is the area of the courtyard outside the fountain?

  74. 74.A window is a rhombus with four equal sides of 7 cm. What is the perimeter of the window?

  75. 75.A triangle has area 16 square inches and a base of 8 inches. The height is perpendicular to that base. What is the height?

  76. 76.An apartment floor joins a 12 foot by 5 foot rectangular room to a 4 foot by 3 foot rectangular room. The rooms do not overlap. What is the combined floor area?

  77. 77.A garden plot is a trapezoid. The parallel sides are 10 yards and 6 yards, and the perpendicular height between them is 4 yards. What is the area of the plot?

  78. 78.A wall mural has an area of 48 square feet. A similar sketch is drawn at a scale factor of 1/2. What is the area of the sketch?

  79. 79.A shipping box is a rectangular prism 4 inches long, 5 inches wide, and 6 inches high. What is the volume of the box?

  80. 80.A drawer is a rectangular prism 8 cm long, 3 cm wide, and 2 cm high. What is the volume of the drawer?

  81. 81.A planter is a rectangular prism with a base of 9 feet by 4 feet. The planter has a volume of 180 cubic feet. What is the height?

  82. 82.A water tank is a cylinder with radius 4 feet and height 10 feet. What is the volume of the tank?

  83. 83.A soup can is a cylinder with radius 5 cm and height 6 cm. What is the volume of the can?

  84. 84.A column is a cylinder with diameter 8 meters and height 7 meters. What is the volume of the column?

  85. 85.A concrete step is a rectangular prism 10 inches long, 10 inches wide, and 3 inches high. What is the volume of the step?

  86. 86.A candle is a cylinder with radius 3 inches and height 5 inches. What is the volume of the candle?

  87. 87.A crate is a rectangular prism 6 feet long, 2 feet wide, and 3 feet high. What is the volume of the crate?

  88. 88.A cylindrical barrel has a volume of 100 pi cubic inches and a height of 4 inches. What is the radius of the barrel?

  89. 89.A wooden block is a rectangular prism 7 cm long, 2 cm wide, and 5 cm high. What is the volume of the block?

  90. 90.A drum is a cylinder with radius 6 inches and height 2 inches. What is the volume of the drum?

  91. 91.A rectangular block has a volume of 240 cubic cm. Two edges measure 8 cm and 6 cm. What is the length of the third edge?

  92. 92.A brick is a rectangular prism 9 cm long, 3 cm wide, and 4 cm high. What is the volume of the brick?

  93. 93.A rolling pin is a cylinder with radius 3 inches and height 7 inches. What is the volume of the rolling pin?

  94. 94.A pedestal is a rectangular prism 5 feet long, 5 feet wide, and 4 feet high. What is the volume of the pedestal?

  95. 95.On a coordinate plane, a trail runs in a straight line from (1, 2) to (5, 10). What is the slope of the trail?

  96. 96.On a coordinate plane, a road runs in a straight line from (2, 9) to (6, 1). What is the slope of the road?

  97. 97.On a coordinate plane, point A is at (2, 4) and point B is at (8, 10). What is the midpoint of segment AB?

  98. 98.A map uses a coordinate grid in which each unit is one block. A library is at (1, 2) and a station is at (4, 6). How many blocks long is the straight path between them?

  99. 99.A drone starts at (0, 0) on a coordinate grid and flies in a straight line to (6, 8). Each grid unit is 1 meter. How many meters is the flight?

  100. 100.Line p is drawn on a coordinate plane and has slope 3. Line q is parallel to line p and is a different line. What is the slope of line q?

  101. 101.Line m on a coordinate plane has slope 4. What is the slope of a line perpendicular to line m?

  102. 102.A line on the coordinate plane has slope 2 and y-intercept 5. Which equation represents the line?

  103. 103.A line on the coordinate plane has equation y = -3x + 7. What is the slope of the line?

  104. 104.On a coordinate plane, a line passes through (3, 5) and (8, 5). What is the slope of the line?

  105. 105.On a coordinate plane, a line passes through (4, 1) and (4, 7). What is the slope of the line?

  106. 106.A triangle on the coordinate plane has vertices at (0, 0), (6, 0), and (0, 4). The base lies on the x-axis from (0, 0) to (6, 0), and the height is the vertical side from (0, 0) to (0, 4). What is the area of the triangle?

  107. 107.A rectangle on the coordinate plane has vertices (1, 2), (7, 2), (7, 6), and (1, 6). The horizontal sides run from x = 1 to x = 7, and the vertical sides run from y = 2 to y = 6. What is the area of the rectangle?

  108. 108.A line on the coordinate plane has equation y = 2x - 6. What is the x-intercept of the line?