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MathidiaMathematics Academy
Exact number and rational reasoning
Lesson 14 of 75
Challenge

Exact values

Objective. Choose exact fractions or π rather than premature decimals.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Exact values are answers left as fractions, surds or multiples of π rather than rounded to a decimal, preserving the true value of a calculation. A calculation should stay exact whenever a question asks for an 'exact' answer, or when rounding at an intermediate stage would introduce error into a later step. For example, the circumference of a circle with radius 7 should be written as 14π rather than 43.98, unless a decimal is specifically requested. Fractions like 2/3 should not be converted to 0.666... when an exact value is wanted, since that decimal never terminates. Recognising when to keep π, surds or fractions is essential in multi-step geometry and algebra problems.

Worked example

A circle has radius 5 cm. Give the exact area of the circle in terms of π.

Diagram of a circle
  1. 1The formula for area is A = πr².
  2. 2Substitute r = 5: A = π × 5².
  3. 35² = 25.
  4. 4A = 25π.

Answer: 25π cm²

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Practice

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Common mistake. Pupils often round π to 3.14 automatically, even when a question explicitly asks for an answer in terms of π, losing marks for inexactness. Leave π symbolic unless told to give a decimal.

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