Math 9 · Square roots and surface area
Undoing a square, and wrapping a solid
A square root undoes squaring, and a good share of the no-calculator Part A of the PAT is knowing which numbers undo cleanly and roughly where the others land. The second half of the unit is surface area: add up every face you could paint, then take away the ones hidden where two pieces touch.
- 1. The words, first
- 2. Square roots of fractions and decimals
- 3. Estimating a root that is not exact
- 4. Surface area of a composite object
- 5. What costs marks
The words, first
The idea: A square root is the side of a square whose area you know. Most of this unit follows from that picture.
| Word | What it means |
|---|---|
| Square of a number | The number times itself: 7² = 7 × 7 = 49. It is the area of a square with side 7. |
| Square root | The number that was squared. √49 = 7, because 7² = 49. The √ sign always means the positive root. |
| Perfect square | A number whose square root is exact. 49, 0.49 and 49/100 are perfect squares; 4.9 is not. |
| Benchmark | A perfect square close to the number you are working with, used to estimate its root. For √20 the benchmarks are 16 and 25. |
| Surface area | The total area of every face of a solid — everything you could paint. Always in square units. |
| Right prism, right cylinder | A solid with two identical parallel bases joined by sides standing straight up. A box is a prism; a can is a cylinder. |
| Composite object | A solid built from two or more simpler ones joined together. |
| Overlap | The area where two pieces of a composite object touch. It is hidden, so it is not part of the surface. |
Square roots of fractions and decimals
The idea: A fraction or a decimal is a perfect square when its top and bottom are both perfect squares. Then the root of the whole is the root of each part.
√(9/16) = √9 / √16 = 3/4 · √0.36 = √(36/100) = 6/10 = 0.6
For a decimal, write it as a fraction first. 0.36 is 36 hundredths, and both 36 and 100 are perfect squares.
Worked example. Is 1.69 a perfect square? 1.69 = 169/100, and 169 = 13² and 100 = 10², so yes: √1.69 = 13/10 = 1.3. Check: 1.3 × 1.3 = 1.69 ✓.
The decimal-places shortcut. A square root has half as many decimal places as the number: √0.0049 = 0.07. That is also why √0.4 is not 0.2 — 0.2² is 0.04.
A root can be bigger than the number. For numbers between 0 and 1 it always is: √0.36 = 0.6. Squaring 0.6 multiplies it by something less than 1, which shrinks it to 0.36.
Estimating a root that is not exact
The idea: Find the perfect squares on either side, then ask which one the number is closer to. That places the root to about a tenth, with no calculator.
Worked example: √52.
- Benchmarks. 49 = 7² and 64 = 8², so √52 is between 7 and 8.
- Closer to which? 52 is 3 of the 15 steps from 49 to 64, so √52 is about 3/15 = 0.2 of the way from 7 to 8: about 7.2.
- Check by squaring. 7.2² = 51.84, just under 52 ✓. A calculator gives 7.21.
The same move handles √20: between √16 = 4 and √25 = 5, and 20 is 4 of the 9 steps along, so about 4.4. (It is 4.47.) This method always lands a little low, so the square-it check is worth the ten seconds.
Decimals work the same way. √0.3 sits between √0.25 = 0.5 and √0.36 = 0.6, a little under halfway, so about 0.55. (It is 0.548.)
On the PAT, square roots and estimation are both on the no-calculator Part A. Knowing the squares up to 15² by heart does most of the work.
Surface area of a composite object
The idea: Add the surface areas of the pieces, then subtract the overlap — twice, because it is hidden on both pieces.
Rectangular prism: SA = 2(lw + lh + wh) · Cylinder: SA = 2πr² + 2πrh
A cylinder is two circles plus a curved side that unrolls into a rectangle, 2πr long and h high.
Worked example. A trophy is a block 10 cm by 10 cm by 4 cm with a cylinder, radius 3 cm and height 6 cm, standing on top. What is its total surface area?
- Block: 2(10 × 10 + 10 × 4 + 10 × 4) = 2(180) = 360 cm².
- Cylinder: 2π(3)² + 2π(3)(6) = 18π + 36π = 54π ≈ 169.6 cm².
- Overlap: the cylinder's bottom circle, π(3)² = 9π, is hidden — and so is the patch of block under it. Subtract it twice: 18π ≈ 56.5 cm².
- Total: 360 + 54π − 18π = 360 + 36π ≈ 473.1 cm².
A quick check. The answer is the block plus the cylinder's curved side only, because the cylinder's top circle exactly replaces the patch it covers. Subtracting the overlap once gives 501.4 cm², and forgetting it gives 529.6 cm². Both look reasonable, which is why they make good wrong answers.
Read what is covered. Painting a shed does not include its floor. On the PAT you may use the π key or 3.14; round only at the end. Math 10C adds pyramids, cones and spheres to this same method.
What costs marks
The idea: Two are about decimals and two are about what is hidden.
- √0.4 = 0.2. Square it to check: 0.2² = 0.04.
- Guessing a root without benchmarks. √52 is between 7 and 8 — not 26, which is half of 52.
- Subtracting the overlap once, or not at all.
- Radius for diameter. A cylinder 8 cm across has r = 4 cm.
- The wrong units. Surface area is in cm² or m², never cm.