Area of a Rhombus - Math Steps, Examples & Questions

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Area of a rhombus

Here you will learn about finding the area of a rhombus, including how to find missing dimensions and convert units to find area.

Students first learn how to find the area of quadrilaterals such as rectangles and squares in the $3$ rd grade and continue to build upon that knowledge as they progress into middle school.

What is the area of a rhombus?

The area of a rhombus is the amount of space inside the rhombus. Like with other 2D shapes, the area is measured in units squares $\left(\mathrm{cm}^2, \mathrm{~m}^2, \mathrm{~mm}^2 \text { etc }\right).$ A rhombus is a special type of parallelogram that has four sides of equal length.

Since a rhombus is a parallelogram, you can find the area the same way you find the area of a parallelogram.

${Area} = length \: of \: the \: base \times length \: of \: the \: height$

${A} = b \times h$

For example, let’s find the area of the rhombus with the given dimensions.

The height of the rhombus is $8{~cm}$ and the base of the rhombus is $10{~cm}.$

Using the formula:

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & h=8 \\\\ & b=10 \\\\ & \text {Area }=8 \times 10 \\\\ & \text {Area }=80 \end{aligned}$

The area of the rhombus is $80 \mathrm{~cm}^2.$

Like parallelograms, rhombuses have diagonals that bisect each other. However, the diagonals of a rhombus also form right angles because they are perpendicular.

You can find the area of a rhombus using diagonals.

For example, find the area of the rhombus with the given dimensions.

A rhombus is a parallelogram, so the diagonals are segment bisectors meaning they cut each other into two equal halves.

Each of the four triangles is equal. The base length is $6{~cm},$ and the height length is $8{~cm}.$ This can also be interpreted as the base length as $8{~cm}$ and the height length as $6{~cm}.$

The area of a triangle is $A = \cfrac{1}{2} \times b \times h$

$\begin{aligned} & A=\cfrac{1}{2} \times b \times h \\\\ & b=6 \\\\ & h=8 \\\\ & A=\frac{1}{2} \times 6 \times 8 \\\\ & A=24 \end{aligned}$

The area of one of the triangles is $24.$ So the area of the entire rhombus is, $24+24+24+24=96 \text { or } 4(24)=96$

The area of the rhombus is $96 \mathrm{~cm}^2.$

In high school, you will learn how to find the area of a rhombus using the area of rhombus formula which is also used to find the area of any quadrilateral with perpendicular diagonals.

$\text { Area of rhombus }=\frac{1}{2} \times \text { diagonal } 1 \times \text { diagonal } 2$

What is the area of a rhombus?

Common Core State Standards

How does this relate to $6$ th grade math?

Grade 6 – Geometry (6.G.A.1)
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

How to find the area of a rhombus

In order to find the area of a rhombus:

Identify the given dimensions.
Substitute the given values into the formula.
Write down your final answer, including the units.

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Area of a rhombus examples

Example 1: finding area given the base and height

Find the area of the rhombus below:

Identify the given dimensions.

The height of the rhombus is $4.2 \mathrm{~cm}.$

Since a rhombus has four congruent sides, the base of the rhombus is $6.8 \mathrm{~cm}.$

2Substitute the given values into the formula.

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & \text {Area }=6.8 \times 4.2 \\\\ & \text {Area }=28.56 \end{aligned}$

3Write down your final answer, including the units.

The area of the rhombus is $28.56 \mathrm{~cm}^2.$

Example 2: find the height of the rhombus

The area of the rhombus is $120 \mathrm{~in}^2$ and the side length of the rhombus is $12 \text { inches}.$

Find the height of the rhombus.

Identify the given dimensions.

The area of the rhombus is $120 \mathrm{~in}^2$ and the base of the rhombus is the same as the side length, which is $12 \mathrm{~in}.$

Substitute the given values into the formula.

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & 120=12 \times \text { height } \\\\ & 120=12 h \\\\ & 120 \div 12=12 \div 12 \mathrm{~h} \\\\ & 10=h \end{aligned}$

Write down your final answer, including the units.

The height of the rhombus is $10 \text{ inches}.$

Example 3: area of rhombus using diagonals

Find the area of a rhombus where $AC=4 \mathrm{~m}$ and $BD=12 \mathrm{~m}.$

Identify the given dimensions.

The diagonals of the rhombus are perpendicular (form right angles) and since a rhombus is a parallelogram, the diagonals also bisect each other. So, $AC$ and $BD$ get divided into two equal halves.

$\begin{aligned} & 4 \div 2=2 \\\\ & 12 \div 2=6 \end{aligned}$

Substitute the given values into the formula.

The diagonals of the rhombus create four congruent (equal) triangles that have a height of $2{~m}$ and base of $6{~m}.$ (This can also be interpreted as a height of $6{~m}$ and base of $2{~m}.$

$\text{Area of a triangle } =\cfrac{1}{2} \, \times base \times height$

$\text{Area of triangle }=\cfrac{1}{2} \, \times 6 \times 2$

$\text{Area of triangle }=6$

The area of each triangle is $6 \mathrm{~m}^2.$

Write down your final answer, including the units.

There are four equal triangles with an area of $6 \mathrm{~m}^2.$

So the area of the rhombus is:

$\begin{aligned} & \text {Area of rhombus }=6+6+6+6 \text { or } 4(6) \\\\ & \text {Area of rhombus }=24 \mathrm{~m}^2 \end{aligned}$

In high school, you will learn to use the area formula for a quadrilateral with perpendicular diagonals.

$\begin{aligned} & \text {Area }=\cfrac{1}{2} \times \text { diagonal } 1 \times \text { diagonal } 2 \\\\ & \text {Area }=\cfrac{1}{2} \times 4 \times 12 \\\\ & \text {Area }=24 \mathrm{~m}^2 \end{aligned}$

Example 4: find area convert dimensions

Identify the given dimensions.

The base of the rhombus is $13.2{~cm}.$

The height of the rhombus is $0.114{~m}.$

Notice the dimensions are in different units. Before finding the area, make sure the units of the dimensions are the same.

Changing the height which is in meters to centimeters.

$\begin{aligned} & 0.114 \times 100=11.4 \\\\ & 0.114 \mathrm{~m}=11.4 \mathrm{~cm} \end{aligned}$

Substitute the given values into the formula.

$\text{Area of a rhombus } = base \times height$

$\text{Area of a rhombus } =13.2 \times 11.4$

$\text{Area of a rhombus } =150.48$

Write down your final answer, including the units.

The area of the rhombus is $150.48 \mathrm{~cm}^2.$

Example 5: word problem

Daniel has a rhombus shaped wooden sign for school spirit week that has an area of $840 \mathrm{~in}^2.$ He wants to figure out the length of the shorter diagonal in order to make a wooden rod to hang it on. He calculated longer diagonal to be $80 \text { inches}.$ What is the length of the shorter diagonal?

Identify the given dimensions.

The diagonals of the rhombus are perpendicular to each other and bisect each other (cut each other into two equal halves).

The longer diagonal is $80$ inches, so $80 \div 2=40$

Each half of the longer diagonal is $40$ inches. The diagonals of a rhombus also divide the rhombus into four congruent triangles that have equal areas.

Since the area of the rhombus is $840 \mathrm{~in}^2$ , by dividing the area by $4,$ you can find the area of one of those triangles.

$840 \div 4=210$

$210 \mathrm{~in}^2$ is the area of one of the triangles.

Substitute the given values into the formula.

Use the area of one triangle to find half of the length of the shorter diagonal.

$\begin{aligned} & \text {Area of triangle }=\frac{1}{2} \times b \times h \\\\ & 210=\cfrac{1}{2} \times 40 \times h \\\\ & 210=20 h \\\\ & 210 \div 20=20 \div 20 h \\\\ & 10.5=h \end{aligned}$

Write down your final answer, including the units.

The height of one of the triangles is $10.5{~inches},$ which is half the length of one of the diagonals.

So, the entire length of the shorter diagonal is $2 \times 10.5=21$

The shorter diagonal length is $21{~inches}.$

Daniel needs a $21{~inch}$ length wooden rod.

Example 6: find perimeter of a rhombus given the area

Find the perimeter of the rhombus with an area of $2,150 \mathrm{~cm}^2$ and a height of $43 \mathrm{~cm}.$

Identify the given dimensions.

The height of the rhombus is $43 \mathrm{~cm}$ and the area is $2,150 \mathrm{~cm}^2.$

Substitute the given values into the formula.

Using the formula:

$Area = base \times height$

$\begin{aligned} & 2150=\text { base } \times 43 \\\\ & 2150=b \times 43 \\\\ & 2150 \div 43=b \times 43 \div 43 \\\\ & 50=b \end{aligned}$

Write down your final answer, including the units.

The base of the rhombus is $50{~cm}.$ Since each side of a rhombus is equal in length, each side of the rhombus is $50{~cm}.$

The perimeter can be found by adding up the side lengths.

$\begin{aligned} & \text {Perimeter }=50+50+50+50 \\\\ & \text {Perimeter }=200 \end{aligned}$

The perimeter of the rhombus is $200{~cm}.$

Teaching tips for area of rhombus

Teach the conceptual understanding of formulas, such as $\text { Area }=\text { base} \times \text { height}$ by doing active learning activities in class so that students understand what the formula means.

Infuse learning activities that provide students with an opportunity to explore alternate ways of finding the area of a rhombus, such as dividing it into two or four equal triangles.

Use digital platforms such as Desmos so students can investigate strategies to find the area of a rhombus.

Although worksheets provide an opportunity for students to practice skills, using digital platforms that focus on game play is more effective in engaging students.

Easy mistakes to make

Forgetting that area is a square unit of measurement
After calculating the area of a 2D shape, students often forget to include the square unit. Area is a two-dimensional measurement, for example,

$\begin{aligned} & \text {Area }=(6 \mathrm{~cm}) \times(4 \mathrm{~cm}) \\ & \text {Area }=24 \mathrm{~cm}^2 \end{aligned}$

Forgetting to convert measures to a common unit
Before finding the area of the rhombus or any 2D shape, first make sure all the units are the same. For example, if different units are given $($ For example, length $= 4{~m}$ and width $= 3{~cm})$ , then you must convert them either both to $cm$ or both to $m.$

Forgetting that a rhombus is a special parallelogram
Rhombuses have the properties of parallelograms. Therefore, to find the area of a rhombus you can use the formula for finding the area of a parallelogram, $\text { Area }=\text { base } \times \text { height.}$

Practice area of a rhombus questions

96 \mathrm{~cm}

96 \mathrm{~cm}^2

48 \mathrm{~cm}

48 \mathrm{~cm}^2

A rhombus is a parallelogram so you can use the formula,

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & \text {Area }=12 \times 8 \\\\ & \text {Area }=96 \end{aligned}$

The area of the rhombus is $96 \mathrm{~cm}^2.$

13 \text { inches }

12 \text { inches}{}^2

\text { 13. } 1 \text { inches}

\text { 13. } 1 \text { inches}{}^2

Substitute known values into the equation, $\text { Area }=\text { base } \times \text { height }$

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & 94.32=\text { base } \times 7.2 \\\\ & 94.32=b \times 7.2 \\\\ & 94.32 \div 7.2=b \times 7.2 \div 7.2 \\\\ & 13.1=b \end{aligned}$

The base equals $13.1$ inches which means each side is equal to $13.1$ inches.

192 \mathrm{~m}

192 \mathrm{~m}^2

96 \mathrm{~m}

96 \mathrm{~m}^2

The diagonals of the rhombus are perpendicular and bisect each other (cut each other into two equal halves). The diagonals also divide the rhombus into four equal triangles with equal areas.

Area of a Rhombus image 17 US

$Area \: of \: triangle =\cfrac{1}{2} \times base \times height$

$Area \: of \: triangle =\cfrac{1}{2} \times 6 \times 8$

$Area \: of \: triangle =24 \mathrm{~m}$

If one of the triangles has an area of $24{~m},$ then all four triangles have an area of $24{~m}.$ So the area of the rhombus is $24+24+24+24=96$ or $4(24)=96.$

Area of the rhombus is $96{~m}^2.$

182.4 \mathrm{~cm}^2

1.824 \mathrm{~m}^2

182 \mathrm{~m}

1.824 \mathrm{~cm}^2

The dimensions are given in different units. First, change them to the same unit. There are $100{~cm}$ in $1$ meter so, $0.12 \times 100=12 \mathrm{~cm}.$

Substitute the values into the equation.

$\begin{aligned} & \text {Area }=\text { base } \times \text { height } \\\\ & \text {Area }=12 \times 15.2 \\\\ & \text {Area }=182.4 \end{aligned}$

The area of the rhombus is $182.4 \mathrm{~cm}^2.$

6 \mathrm{~m}

3 \mathrm{~m}

12 \mathrm{~m}

24 \mathrm{~m}

The diagonals of a rhombus are perpendicular and bisect each other. They also create four congruent triangles with the same areas.

Area of a Rhombus image 20 US

Divide the area of the rhombus by $4$ to find the area of one of the triangles, and then, using the area of a triangle, substitute the known values.

$54 \div 4=13.5$

$\text{Area of triangle } =\cfrac{1}{2} \, \times base \times height$

$13.5=\cfrac{1}{2} \times b \times 4.5$

$13.5=b \times 2.25$

$13.5 \div 2.25=b \times 2.25 \div 2.25$

$6=b$

The base of one triangle is $6 \, m$ which is half the length of the diagonal. So, double the length.

$\begin{aligned} & 6 \times 2=12 \\\\ & B D=12 \mathrm{~m} \end{aligned}$

Another strategy would be to use the formula,

$\begin{aligned} & \text {Area }=\cfrac{1}{2} \times \text { diagonal } 1 \times \text { diagonal } 2 \\\\ & 54=\cfrac{1}{2} \times 9 \times \text { diagonal } 2 \\\\ & 54=4.5 \times \text { diagonal } 2 \\\\ & 12=\text { diagonal } 2 \\\\ & B D=12 \mathrm{~m} \end{aligned}$

56 \mathrm{~cm}

56 \mathrm{~cm}^2

14 \mathrm{~cm}

14 \mathrm{~cm}^2

Use the formula $\text { Area }=\text { base } \times \text { height }$ and substitute the known values.

$\begin{aligned} & 112=\text { base } \times 8 \\\\ & 112=b \times 8 \end{aligned}$

$\begin{aligned} & 112 \div 8=b \times 8 \div 8 \\\\ & 14=b \end{aligned}$

The base of the rhombus is $14{~cm}$ which means the side lengths are $14{~cm}.$

$\begin{aligned} & \text {Perimeter }=14+14+14+14 \text { or } 4(14) \\\\ & \text {Perimeter }=56 \mathrm{~cm} \end{aligned}$

Area of rhombus FAQs

Do you have to use the Pythagorean theorem or trigonometry to find the area of a rhombus?

In high school, you will explore other strategies to find the area of rhombuses that focus on the diagonals creating four smaller right-angled triangles. Since you will be working with right triangles, you will most likely be applying the Pythagorean Theorem or right triangle trigonometry to find the length of the side or the length of the hypotenuse.

Do rhombuses have the same properties as parallelograms?

Yes, rhombuses are parallelograms, so they have opposite sides that are parallel and congruent, opposite angles that are congruent, adjacent angles that are supplementary, and diagonals that bisect each other. The interior angles add up to be $360$ degrees.

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