Keep fractions and multiples of pi exact

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Keep fractions and multiples of pi exact

An exact expression retains the full value. A fraction such as 1/3 is exact, while 0.333 is a rounded approximation. Similarly, π is an exact symbol for the circle constant; 3.14 is only an approximation.

Combine exact fractions with common denominators and simplify products by cancelling factors. Combine multiples of π as like terms: 2π + 3π = 5π. In multiplication, (2π)(3π) = 6π². These different operations must not be conflated.

In a circle calculation, keep π until a decimal answer is requested. For radius 3 cm, circumference is 6π cm and area is 9π cm². The powers and units distinguish a length from an area. A rounded approximation should use ≈ rather than an equality claiming exactness.

Worked example: Find the exact area of a circle with radius 3 cm.

Use area = πr².

Square 3 before multiplying by π.

Attach square-centimetre units.

Answer: 9π cm².

Check your understanding: Is π exactly 3.14?

Answer: No.

Why: 3.14 is a rounded decimal; replacing π changes the exact value.