How to find the exact perimeter of a rectangle with surds

Key takeaway: Exact answers keep the surd form instead of turning everything into decimals. That is what makes this kind of perimeter question neat.

For a rectangle, the perimeter is

$P=2(\text{length}+\text{width}).$

Here the sides are $(2+\sqrt{10})$ cm and $\sqrt{5}$ cm.

So

$P=2\big((2+\sqrt{10})+\sqrt{5}\big).$

Combine inside the bracket:

$P=2(2+\sqrt{10}+\sqrt{5}).$

Distribute the 2:

$P=4+2\sqrt{10}+2\sqrt{5}.$

Exact perimeter

$\boxed{4+2\sqrt{10}+2\sqrt{5}\text{ cm}}$

You can also write it as

$\boxed{4+2(\sqrt{10}+\sqrt{5})\text{ cm}}.$

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Pitfalls the pros know 👇 Don’t add the roots as if $\sqrt{10}+\sqrt{5}=\sqrt{15}$. That is not valid. Also, remember perimeter means all four sides, so for a rectangle you must double the sum of one length and one width.

What if the problem changes? If the sides were both surds, the same formula still works. For example, with sides $\sqrt{2}$ and $\sqrt{8}$, you would get $P=2(\sqrt{2}+\sqrt{8})=2(\sqrt{2}+2\sqrt{2})=6\sqrt{2}$. If one side changes from $(2+\sqrt{10})$ to $(3+\sqrt{10})$, the perimeter just increases by 2 cm.

Tags: perimeter formula, rectangle, exact value