Mathematics • Year 10 • Unit 1 • Lesson 14
Surds in the Real World
Apply Lesson 14's surd operations to the geometry and engineering contexts where exact values matter: square diagonals, special right triangles (1 : √3 : 2), Pythagoras with surd answers, and AC peak-vs-RMS voltage. Then explain in your own words why engineers keep surd form until the very last step.
1. Word problems
Each problem uses surd simplification, like-surd combining, or rationalising — and the answer should stay in exact form unless told otherwise.
1.1 — Side of a square paver. A square concrete paver has area 50 cm². The lesson's SAQ 3 explores this exact scenario.
(a) Find the exact side length in simplest surd form.
(b) Find the exact diagonal length in simplest surd form. (Hint: diagonal = side × √2.)
(c) Explain in one sentence why a paving contractor cutting many of these pavers would use the surd form rather than the decimal approximation. 3 marks
1.2 — A 30°-60°-90° set-square triangle. The lesson's Real-Life Link cites the side-ratio 1 : √3 : 2 for these triangles. An architect's set-square has its shortest side 6 cm long.
(a) State the lengths of the other two sides in exact form (using a surd where needed).
(b) Calculate the perimeter in simplest surd form.
(c) Calculate the area in exact form. 3 marks
1.3 — Pythagoras with a surd hypotenuse. A roof rafter forms a right-angled triangle with legs 4 m and 6 m. Find the exact length of the rafter in simplest surd form.
(a) Apply Pythagoras: c² = a² + b².
(b) Simplify the surd answer.
(c) Compute the decimal value to 2 decimal places — but state which form a carpenter on site would actually cut to. 3 marks
1.4 — AC voltage peak vs RMS. Lesson 14's Real-Life Link cites that the relationship between peak and RMS voltage in an AC circuit involves √2: V_peak = V_RMS × √2. A standard Australian power point delivers V_RMS = 230 V.
(a) Write V_peak in exact surd form.
(b) Calculate the decimal value to the nearest volt.
(c) Conversely, a peak voltage of 100√2 V corresponds to what V_RMS in exact form? 3 marks
1.5 — Combining surds in a perimeter. A rectangular garden has length 5√2 m and width 3√2 m.
(a) Calculate the perimeter in simplest surd form.
(b) Calculate the area in exact form (you should get a whole number).
(c) Compare the area calculation method to what would happen if you used decimal approximations of √2 first. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "Why bother with surds? My calculator gives √2 = 1.41421356, so just use that everywhere." Using everything from Lesson 14, explain (i) what is lost when you replace √2 with a decimal approximation, (ii) why squaring 1.414 does NOT give exactly 2 (test this on your calculator) — and what this tells you about compounding error, and (iii) why architects working with 30°-60°-90° triangles or engineers working with AC voltages keep surd form until the final step. Refer to "exact value" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Square paver
(a) Side = √50 = √(25 × 2) = 5√2 cm.
(b) Diagonal = 5√2 × √2 = 5 × 2 = 10 cm exactly.
(c) The exact form means every paver's diagonal is exactly 10 cm — no compounding rounding errors across dozens of cuts. A decimal approximation of 5√2 ≈ 7.07 cm would propagate small errors into the diagonal calculation.
1.2 — 30°-60°-90° set-square
(a) Sides (in the 1 : √3 : 2 ratio scaled by 6): 6, 6√3, 12 cm.
(b) Perimeter = 6 + 6√3 + 12 = (18 + 6√3) cm.
(c) Area = ½ × 6 × 6√3 = 18√3 cm².
The lesson's Real-Life Link names the 1 : √3 : 2 ratio specifically.
1.3 — Roof rafter
(a) c² = 4² + 6² = 16 + 36 = 52. So c = √52.
(b) 52 = 4 × 13. √52 = √4 × √13 = 2√13 m.
(c) Decimal: 2√13 ≈ 2 × 3.606 ≈ 7.21 m. A carpenter on site would cut to the decimal (with a tape measure marked in cm), but a structural engineer would calculate using 2√13 to keep precision in subsequent load calculations.
1.4 — Australian power point
(a) V_peak = 230 × √2 = 230√2 V exactly.
(b) 230√2 ≈ 230 × 1.4142 ≈ 325 V.
(c) V_RMS = (100√2) / √2 = 100 V. (The √2's cancel cleanly — no need to rationalise here because they appear in both numerator and denominator.)
1.5 — Rectangular garden in surds
(a) Perimeter = 2(5√2 + 3√2) = 2(8√2) = 16√2 m.
(b) Area = (5√2)(3√2) = 5 × 3 × (√2)² = 15 × 2 = 30 m² (whole number, exact).
(c) Using decimals: (5 × 1.414)(3 × 1.414) ≈ 7.07 × 4.243 ≈ 29.99 — close to 30 but no longer exact. Surd form preserves the answer perfectly because (√2)² collapses exactly to 2.
2.1 — Explain your thinking (sample response)
(i) When you replace √2 with 1.41421356 you lose exactness: the decimal terminates somewhere, but √2 has an infinite, non-repeating expansion, so any decimal cuts off real information. (ii) Squaring 1.414 on a calculator gives 1.999396, not 2 — the small rounding error in the decimal got amplified by squaring, and in a longer chain of multiplications the error grows further. (iii) Architects designing 30°-60°-90° trusses, and electrical engineers converting between V_peak and V_RMS in AC circuits, keep the surd until the last step because the exact value protects against this compounding error; only at the very end (when ordering a length of timber or specifying a wire gauge) do they convert to a decimal. That way the geometry stays mathematically perfect and the only rounding happens once.
Marking: 1 for naming what is lost (exactness / infinite non-repeating decimal), 1 for showing that (1.414)² ≠ 2 demonstrates compounding error, 1 for a real-world example (architect or engineer), 1 for using "exact value" correctly.