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ISEE Upper Level Math : Rhombuses

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Example Questions

Example Question #1 : How To Find The Area Of A Rhombus

Two diagonals of a rhombus have lengths of (t+1) and (t-1) . Give the area in terms of .

Possible Answers:

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

t^2+1

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

t^2-1

t^2

Correct answer:

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

Explanation:

The formula for the area of a rhombus is

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

where $d_{1}$ is the length of one diagonal and $d_{2}$ is the length of the other diagonal.

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

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Example Question #1 : Rhombuses

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

Possible Answers:

$6\ cm^2$

$12\ cm^2$

$8\ cm^2$

$10\ cm^2$

$9\ cm^2$

Correct answer:

$8\ cm^2$

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.

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

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Example Question #3 : How To Find The Area Of A Rhombus

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.

Possible Answers:

$288\text{in}^2$

$121\text{in}^2$

$100\text{in}^2$

$48\text{in}^2$

$144\text{in}^2$

Correct answer:

$144\text{in}^2$

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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Example Question #1 : How To Find The Length Of The Side Of A Rhombus

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

Possible Answers:

$9\ in$

$8\ in$

$10\ in$

$12\ in$

$6\ in$

Correct answer:

$9\ in$

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.

Perimeter=4a ,

where is the length of each side.

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

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Example Question #1 : How To Find The Length Of The Side Of A Rhombus

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.

Possible Answers:

$3\ cm$

$2\ cm$

$2\sqrt{2}\ cm$

$3\sqrt{2}\ cm$

$3\sqrt{3}\ cm$

Correct answer:

$3\sqrt{2}\ cm$

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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Example Question #1 : How To Find The Perimeter Of A Rhombus

One side of a rhombus measures 2 yards. Which of the following is equal to the perimeter?

Possible Answers:

$12 \textrm{ yd}$

$18 \textrm{ ft}$

All of the other choices give the correct perimeter.

$\frac{1}{200} \textrm{ mi}$

$288 \textrm{ in }$

Correct answer:

$288 \textrm{ in }$

Explanation:

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength 2 yards is $2 \times 4 = 8$ yards.

$8 \textrm{ yd} = 8 \times 3 = 24 \textrm{ ft} = 24 \times 12 = 288 \textrm{ in}$

Also,

$8 \textrm{ yd} = 24 \textrm{ ft} =\frac{ 24}{5,280} = \frac{1}{220} \textrm{ mi}$

The only choice that agrees with any of these measures is 288 inches, the correct choice.

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Example Question #1 : Rhombuses

One side of a rhombus measures y + 12 . Give its perimeter in terms of .

Possible Answers:

$y^{2}+ 24y + 144$

3y + 36

$y^{2}+ 12y + 144$

2y + 24

4y + 48

Correct answer:

4y + 48

Explanation:

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength y + 12 is 4 (y + 12) = 4y + 48 .

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Example Question #1 : How To Find An Angle In A Rhombus

Consider the rhombus below. Solve for .

Problem_10

Possible Answers:

x=35

x=65

x=325

x=145

Correct answer:

x=145

Explanation:

The total sum of the interior angles of a quadrilateral is 360 degrees. In this problem, we are only considering half of the interior angles:

$\frac{360}{2}=180$

35+x=180

x=145

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Example Question #2 : How To Find An Angle In A Rhombus

Rhombus

Note: Figure NOT drawn to scale.

The above depicts a rhombus and one of its diagonals. What is ?

Possible Answers:

x = 48

x = 42

x = 58

x = 29

x = 32

Correct answer:

x = 29

Explanation:

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\left (180 - 122 \right ) ^{\circ } = 58^{\circ }$ , and is half that, or $29^{\circ }$ .

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Rhombuses

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Area Of A Rhombus

Example Question #1 : Rhombuses

Example Question #3 : How To Find The Area Of A Rhombus

Example Question #1 : How To Find The Length Of The Side Of A Rhombus

Example Question #1 : How To Find The Length Of The Side Of A Rhombus

Example Question #1 : How To Find The Perimeter Of A Rhombus

Example Question #1 : Rhombuses

Example Question #1 : How To Find An Angle In A Rhombus

Example Question #2 : How To Find An Angle In A Rhombus

All ISEE Upper Level Math Resources

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