How to find the length of the side of a rhombus

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ISEE Upper Level Quantitative Reasoning › How to find the length of the side of a rhombus

Questions 1 - 2

1

A rhombus has a perimeter of $36\ in$ . Give the side length of the rhombus.

$9\ in$

$8\ in$

$10\ in$

$12\ in$

$6\ in$

Explanation

Like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be found by adding together the length of each side. In the case of a rhombus, all four sides have the same length, i.e.

,

where is the length of each side.

$Perimeter=4a\Rightarrow 36=4a\Rightarrow a=9\ in$

2

A rhombus has the area of $9\sqrt{2}\ cm^2$ . One of the interior angles is $45^{\circ}$ . Give the length of each side of the rhombus.

$3\sqrt{2}\ cm$

$2\sqrt{2}\ cm$

$3\sqrt{3}\ cm$

$3\ cm$

$2\ cm$

Explanation

The area of a rhombus can be determined by the following formula:

$Area=s^2sin\alpha$ ,

where is the length of any side and $\alpha$ is any interior angle.

$Area=s^2sin\alpha\Rightarrow 9\sqrt{2}=s^2\times sin45^{\circ}\Rightarrow 9\sqrt{2}=s^2 \times \frac{\sqrt{2}}{2}$

$\Rightarrow s^2=18\Rightarrow s=\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}\ cm$

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