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Math 10C

Math 10C · Worksheets

Mixed review · Measurement and trigonometry

Ten questions of mixed difficulty, covering Measurement, Surface area and volume, Trigonometry. Print it, or work through it on screen — the answer key starts on its own page.

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Mixed review · Measurement and trigonometry

Math 10C · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. In a right triangle the side opposite θ is 5 and the hypotenuse is 13. Then sin θ is

    1. a) 13/5
    2. b) 5/13
    3. c) 5/12
    4. d) 12/13

  2. In a right triangle, the hypotenuse is

    1. a) the side opposite the angle you are using
    2. b) the side opposite the right angle, and always the longest
    3. c) always the vertical side
    4. d) the side next to the angle you are using

  3. Compared with a cylinder of the same radius and height, a cone has

    1. a) three times the volume
    2. b) half the volume
    3. c) the same volume
    4. d) one third of the volume

  4. To find a missing angle when you know two sides, you use

    1. a) the sum of angles only
    2. b) the Pythagorean theorem
    3. c) a regular trig function
    4. d) an inverse trig function, such as sin⁻¹

  5. The volume of a cylinder is

    1. a) πr²h
    2. b) πr²
    3. c) ⅓πr²h
    4. d) 2πrh

  6. A recipe needs 2 cups of liquid and 1 cup is 250 mL. In litres that is

    1. a) 0.5 L
    2. b) 500 L
    3. c) 0.25 L
    4. d) 2.5 L

  7. A solid is a cylinder of radius 2 cm and height 8 cm with a hemisphere of the same radius on top. Its volume is about

    1. a) 101 cm³
    2. b) 134 cm³
    3. c) 117 cm³
    4. d) 84 cm³

  8. A map has a scale of 1 : 50 000. A distance of 6 cm on the map represents

    1. a) 300 m
    2. b) 500 m
    3. c) 3 km
    4. d) 30 km

  9. A ladder 6 m long leans against a wall and reaches 5 m up it. The angle it makes with the ground is about

    1. a) 50°
    2. b) 40°
    3. c) 34°
    4. d) 56°

  10. A square-based pyramid has base edge 6 cm and height 10 cm. Its volume is

    1. a) 120 cm³
    2. b) 360 cm³
    3. c) 200 cm³
    4. d) 60 cm³

Answer key · Mixed review · Measurement and trigonometry

Math 10C · maddyhelps.com

  1. b) 5/13 — Sine is opposite over hypotenuse, so 5/13. The third side is 12 by Pythagoras, which would give cos θ = 12/13 and tan θ = 5/12 — but the question asked only for sine.
  2. b) the side opposite the right angle, and always the longest — The hypotenuse never changes as you switch which acute angle you are working from — it is fixed by the right angle. The opposite and adjacent sides do swap over, which is why labelling has to be done for one angle at a time.
  3. d) one third of the volume — Cones and pyramids both get a third of the prism or cylinder they fit inside. It is worth remembering as one fact covering two shapes rather than two separate formulas.
  4. d) an inverse trig function, such as sin⁻¹ — sin θ gives a ratio from an angle; sin⁻¹ gives an angle from a ratio. On a calculator it is usually the second function of the sine key, and forgetting to use it returns a nonsensical number.
  5. a) πr²h — A cylinder is a prism with a circular base, so it is base area times height, and the base area is πr². The second option is the curved surface area and the fourth is the volume of a cone.
  6. a) 0.5 L — 2 × 250 = 500 mL, and 1000 mL make a litre, so 0.5 L. Two conversions in a row: first to the unit you were given a factor for, then to the one asked for.
  7. c) 117 cm³ — Add the parts: cylinder π(4)(8) ≈ 100.5, hemisphere ½ × (4/3)π(8) ≈ 16.8, total ≈ 117 cm³. For a composite <i>volume</i> you always add; for a composite <i>surface area</i> you also have to remove the faces that are now joined.
  8. c) 3 km — 6 cm × 50 000 = 300 000 cm. Then 300 000 ÷ 100 = 3000 m = 3 km. A scale is a ratio of the same units, so convert only after multiplying.
  9. d) 56° — The height is opposite the angle and the ladder is the hypotenuse, so sin θ = 5/6 and θ = sin⁻¹(0.833) ≈ 56°. Drawing the triangle and labelling the sides before choosing the ratio is the whole method.
  10. a) 120 cm³ — V = ⅓ × base area × height = ⅓ × 36 × 10 = 120 cm³. The base area uses the base edge, not the slant height — those are different lengths and questions often give both.