Exact lengths, areas and volumes with surds

When the measurements of a shape are given as surds, the formulae for area, volume and Pythagoras' theorem still apply. Keep every quantity exact: unless a question asks for a decimal answer, the answer is expected in a stated surd form such as \(a+b\sqrt3\), worked without a calculator. When a decimal is asked for, find the exact value first and round only at the end.

Why: a formula such as area \(=\) length \(\times\) width only multiplies and adds numbers, and surds obey the same rules. An unknown length is found by dividing, which leaves a surd in the denominator; rationalising removes it.

Method

1. Write the formula with the unknown as the subject, for example length \(=\dfrac{\text{area}}{\text{width}}\) or \(h=\dfrac{V}{\pi r^2}\).
2. Square or expand any surd lengths first, for example \(r^2=(1+\sqrt5)^2=6+2\sqrt5\).
3. Divide by multiplying the numerator and the denominator by the conjugate of the denominator.
4. Simplify to the form asked, state the units, and check that a length is positive.

In a cuboid with edges \(a\), \(b\) and \(c\), the longest straight line inside it is the space diagonal \(d\). Use Pythagoras' theorem twice: across the base, the square of the base diagonal is \(a^2+b^2\); that diagonal is at right angles to the vertical edge \(c\), so \(d^2=a^2+b^2+c^2\).

Example

A rectangle has area \((10+4\sqrt3)\) cm\(^2\) and width \((2+\sqrt3)\) cm. Find its length and the square of its diagonal.

\(\begin{aligned} \text{length}&=\dfrac{10+4\sqrt3}{2+\sqrt3}\times\dfrac{2-\sqrt3}{2-\sqrt3} \\ &=\dfrac{20-10\sqrt3+8\sqrt3-12}{4-3} \\ &=8-2\sqrt3 \end{aligned}\)

The length is \((8-2\sqrt3)\) cm, which is positive because \(2\sqrt3=\sqrt{12}<4\).

\(\begin{aligned} d^2&=(8-2\sqrt3)^2+(2+\sqrt3)^2 \\ &=(64-32\sqrt3+12)+(4+4\sqrt3+3) \\ &=83-28\sqrt3 \end{aligned}\)

The square of the diagonal is \((83-28\sqrt3)\) cm\(^2\).