How to find the area of a rhombus

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

Questions 1 - 3

1

Two diagonals of a rhombus have lengths of and . Give the area in terms of .

$\frac{t^2-1}{2}$

$\frac{t^2+1}{2}$

Explanation

The formula for the area of a rhombus is

$Area=\frac{d_{1}d_{2}}{2}$ ,

where is the length of one diagonal and is the length of the other diagonal.

$Area=\frac{d_{1}d_{2}}{2}=\frac{(t-1)(t+1)}{2}=\frac{t^2-1}{2}$

2

A rhombus has side length $4\ cm$ , and one of the interior angles is $30^{\circ}$ . Give the area of the rhombus.

$8\ cm^2$

$10\ cm^2$

$6\ cm^2$

$12\ cm^2$

$9\ cm^2$

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.

$Area=s^2sin\alpha=4^2\times sin30^{\circ}=16\times \frac{1}{2}=8\ cm^2$

3

Find the area of a rhombus with one diagonal having a length of 12in and the other having a length two times the first diagonal.

$144\text{in}^2$

$288\text{in}^2$

$48\text{in}^2$

$121\text{in}^2$

$100\text{in}^2$

Explanation

To find the area of a rhombus, we will use the following formula:

$A = \frac{pq}{2}$

where p and q are the lengths of the diagonals of the rhombus.

Now, we know one diagonal has a length of 12in. We also know the other diagonal is two times the first diagonal. Therefore, the second diagonal is 24in.

Knowing this, we can substitute into the formula. We get

$A = \frac{12\text{in} \cdot 24\text{in}}{2}$

$A = \frac{288\text{in}^2}{2}$

$A = 144\text{in}^2$

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