An exact answer preserves a result without rounding. In GCSE maths, exact answers take three forms: a simplified surd (such as 3√2), a fraction (such as 5/3), or an expression involving π (such as 6π). Knowing when to leave an answer in exact form — and how to write it — earns marks across many topics.
What does "exact form" mean in GCSE maths?
A decimal approximation like 4.472 loses information because it has been rounded. An exact form represents the same value with no rounding error. The three types you will meet at GCSE are:
- Surds: √2, √5, 3√3, 2 − √7. Used in geometry (Pythagoras, trigonometry) and algebra.
- Fractions: ⅔, 7/12. Used when the result of a calculation is not an integer.
- Multiples of π: 6π, 4π/3, 25π/2. Used in circle calculations (area, circumference, arc length, volume of sphere/cylinder/cone).
How do you spot when a question wants an exact answer?
Look for these signals in the wording:
| Phrase | What it means |
|---|---|
| "Give your answer as a surd" | Leave in simplified surd form |
| "Give your answer in terms of π" | Leave π as a symbol, do not use 3.14… |
| "Give your answer as a fraction" | Do not convert to a decimal |
| "Show that…" or "Prove that…" | Work with exact values throughout |
| "Give your answer in exact form" | Any of the above, as appropriate |
| No instruction given | Use the context — non-calculator paper often expects exact form |
If a non-calculator paper question involves √2 or π and no rounding instruction is given, exact form is almost certainly expected.
How do you leave Pythagoras answers in surd form?
When the square root of the answer is not a perfect square, simplify the surd rather than evaluating it as a decimal.
Worked example 1: Pythagoras with a surd answer
A right-angled triangle has legs 5 cm and 7 cm. Find the hypotenuse exactly.
h² = 5² + 7² = 25 + 49 = 74
h = √74
Can √74 be simplified? 74 = 2 × 37. Neither factor is a perfect square. So the exact answer is √74 cm.
Worked example 2: surd that simplifies
A right-angled triangle has legs 4 cm and 6 cm. Find the hypotenuse.
h² = 16 + 36 = 52
h = √52 = √(4 × 13) = 2√13 cm.
Exact answer: 2√13 cm
How do you leave circle answers in terms of π?
Substitute all values and simplify, leaving π as a symbol.
Worked example 3: area of a circle
A circle has radius 7 cm. Find the exact area.
Area = πr² = π × 7² = 49π cm²
Do not write 153.9… unless the question specifically says "to 3 s.f." or similar.
Worked example 4: volume of a cylinder
A cylinder has radius 3 cm and height 8 cm.
Volume = πr²h = π × 9 × 8 = 72π cm³
How do you leave trigonometric answers in exact form?
For the exact values of sin, cos, and tan at 30°, 45°, and 60°, use the table rather than a decimal.
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | ½ | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
Worked example 5
A right-angled triangle has hypotenuse 10 cm and an angle of 30°. Find the exact length of the opposite side.
sin 30° = opposite ÷ 10
opposite = 10 × sin 30° = 10 × ½ = 5 cm (this is exact and also an integer).
Worked example 6
Hypotenuse 8 cm, angle 45°. Find the adjacent side exactly.
cos 45° = adjacent ÷ 8
adjacent = 8 × √2/2 = 4√2 cm
Frequently asked questions
Can I always leave an answer as a surd, even if the question doesn't ask?
If the paper says "calculator allowed" and the question doesn't specify, round to a suitable degree of accuracy (usually 3 s.f. or as instructed). On a non-calculator paper without rounding guidance, exact form is generally preferred. When in doubt, give both the exact form and the decimal — this covers all mark schemes.
What if my exact answer looks complicated, like (3 + √5)/2?
Leave it exactly as it is. An exact fractional surd is still exact. Do not "tidy" it into a decimal — that defeats the purpose. Check that any common factor has been cancelled from the fraction and that the surd is fully simplified.
Why does GCSE ask for exact answers?
Exact answers avoid accumulated rounding errors in multi-step calculations. They also signal a deeper understanding — you know the mathematical structure of the result, not just its approximate size. In certain proofs and "show that" questions, only exact working is accepted.
Does leaving in terms of π mean I treat π as a number?
π is a specific irrational number (approximately 3.14159…). "Leave in terms of π" means you write, say, 49π rather than calculating 49 × 3.14159 = 153.938…. You can still simplify the coefficient: 24π/6 = 4π. Treat π like an unknown in algebra — simplify the number part, leave π alone.
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