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Square roots and surface area

Ten questions of mixed difficulty, covering Square roots. Print it, or work through it on screen — the answer key starts on its own page.

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Square roots and surface area

Math 9 · maddyhelps.com

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

  1. √40 lies between which two whole numbers?

    1. a) 7 and 8
    2. b) 6 and 7
    3. c) 5 and 6
    4. d) 20 and 21

  2. √(9/16) equals

    1. a) 9/32
    2. b) 9/4
    3. c) 3/4
    4. d) 3/16

  3. √0.36 equals

    1. a) 0.06
    2. b) 0.18
    3. c) 0.6
    4. d) 6

  4. Without a calculator, the best estimate of √27 is

    1. a) 4.8
    2. b) 5.2
    3. c) 13.5
    4. d) 5.7

  5. The whole number closest to √85 is

    1. a) 8
    2. b) 43
    3. c) 9
    4. d) 10

  6. A cube with 10 cm edges sits in the centre of the top of a rectangular prism that is 20 cm long, 20 cm wide and 5 cm high. What is the surface area of the object?

    1. a) 1300 cm²
    2. b) 1800 cm²
    3. c) 1700 cm²
    4. d) 1600 cm²

  7. A square garden has an area of 2.25 m². How much fencing is needed to go all the way around it?

    1. a) 3 m
    2. b) 6 m
    3. c) 9 m
    4. d) 1.5 m

  8. A cylinder with radius 2 cm and height 5 cm stands on top of a rectangular prism that is 8 cm by 8 cm by 3 cm. To the nearest square centimetre, what is the surface area of the object?

    1. a) 350 cm²
    2. b) 312 cm²
    3. c) 287 cm²
    4. d) 299 cm²

  9. Which number has a square root between 0.7 and 0.8?

    1. a) 6.4
    2. b) 0.75
    3. c) 0.6
    4. d) 0.06

  10. Two cube-shaped boxes, each with 30 cm edges, are taped together face to face. What is the surface area of the pair as one object?

    1. a) 5400 cm²
    2. b) 10 800 cm²
    3. c) 9900 cm²
    4. d) 9000 cm²

Answer key · Square roots and surface area

Math 9 · maddyhelps.com

  1. b) 6 and 7 — Find the perfect squares on either side: 36 < 40 < 49, so √40 is between √36 = 6 and √49 = 7. Halving 40 to get 20 is the most common way to get this wrong.
  2. c) 3/4 — Take the root of the top and the root of the bottom: √9 = 3 and √16 = 4, so 3/4. Rooting only one of them gives 3/16 or 9/4. Check by squaring: (3/4)² = 9/16 ✓.
  3. c) 0.6 — 0.6 × 0.6 = 0.36. A square with two decimal places has a root with one, so 0.06 is too small — it is the root of 0.0036. And 0.18 is 0.36 halved: a square root is not division by 2.
  4. b) 5.2 — 27 is 2 more than 25 but 9 less than 36, so √27 is only a little more than 5 — about 5.2. Guessing near the halfway point ignores how much closer 27 is to 25, and 4.8 is on the wrong side of 5.
  5. c) 9 — 85 is between 81 = 9² and 100 = 10², and it is only 4 away from 81 but 15 away from 100, so √85 is just over 9 (about 9.2). 43 comes from halving 85, which is not what a square root does.
  6. d) 1600 cm² — Cube 6 × 100 = 600 cm², prism 2(400 + 100 + 100) = 1200 cm², and the 100 cm² where they meet is hidden on both: 1800 − 200 = 1600 cm². Only the part of the prism's top under the cube is hidden — removing the whole 400 cm² top gives 1300.
  7. b) 6 m — Each side is √2.25 = 1.5 m, because 1.5 × 1.5 = 2.25, and a square has four sides: 4 × 1.5 = 6 m. Stopping at 1.5 m gives one side, and 9 m comes from multiplying the area by 4 without taking the root first.
  8. c) 287 cm² — Prism 2(64 + 24 + 24) = 224 cm²; cylinder 2πr² + 2πrh = 8π + 20π ≈ 88 cm². The circle where they touch is hidden on the cylinder's base and on the prism's top, so subtract 2 × 4π ≈ 25 cm²: 224 + 88 − 25 ≈ 287 cm². Ignoring the overlap gives 312, and subtracting the circle once gives 299.
  9. c) 0.6 — Square the ends: 0.7² = 0.49 and 0.8² = 0.64, so the number is between 0.49 and 0.64, and 0.6 is (√0.6 ≈ 0.77). Choosing 0.75 forgets to square — its root is about 0.87 — and 6.4 comes from writing 0.8² as 6.4 instead of 0.64.
  10. d) 9000 cm² — Each face is 30 × 30 = 900 cm², and two cubes have 12 faces: 10 800 cm². Taping them hides one face on each box, so subtract 2 × 900 to get 9000 cm². Subtracting only one face gives 9900.