3.4Approximation Errors

When calculations have multiple steps, rounding too early can cause errors to accumulate. These are called premature rounding errors.

The area of a circle is 54 cm². Find its circumference, correct to 3 s.f.

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A = \pi r^2 = 54

, so

r = \sqrt{\frac{54}{\pi}} = 4.14593\ldots

Keep that in the calculator. Circumference

= 2\pi r = 26.0

cm (to 3 s.f.)

Now round

r

4.15

first:

2\pi \times 4.15 = 26.07\ldots

, which gives

26.1

cm.

26.0

and

26.1

are different answers, and only the first is right.

Round once, at the end, to the accuracy the question asks for.

The area of a square is given as $850$ cm² correct to 2 significant figures. Find the range of possible perimeters.

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Since

850

cm² is correct to 2 s.f., the area

A

satisfies

845 \leq A < 855

The side length

s = \sqrt{A}

, so

s

ranges from

\sqrt{845}

\sqrt{855}

\sqrt{845} = 29.0688\ldots

cm and

\sqrt{855} = 29.2403\ldots

cm.

Perimeter

= 4s

, so the perimeter

P

satisfies

4\sqrt{845} \leq P < 4\sqrt{855}

4\sqrt{845} = 116.2755\ldots

cm and

4\sqrt{855} = 116.9615\ldots

cm.

Therefore

116.27 \text{ cm} \leq P < 116.97 \text{ cm}

, rounding each bound outwards so that no possible perimeter is left out.

Approximation and Estimation