The Area of a Rhombus

The first formula to calculate the area of a rhombus.

$Area=\frac{Diagonal_1\times Diagonal_2}{2}$

The second formula to calculate the area of a rhombus.

$Area=Height\times Side$

Below is a variety of exercises to calculate the area of a rhombus:

• Diagonal $a=8$
• Diagonal $b=13$

Question:

What is the area of the rhombus?

Explanation

The calculation is done as follows: $13\times8=104$. Now, the amount should be divided by $2$, and then the obtained answer is $52$. So, you will calculate the area of the rhombus according to the data presented to you in the question.

Answer:

$Area = 52$

Diagonal $a=2$
Diagonal $b=5$

Question:

What is the area of the rhombus?

Explanation
The area of the rhombus $=2\times5=10$.

Now we divide this result by $2$

Answer

The area of the rhombus is $5$.

Diagonal $a=6$
Diagonal $b=4$

Question:

What is the area of the rhombus?

Explanation
The area of the rhombus $=6\times4=24$.

Then we divide by $2$

Answer

The area of the rhombus is therefore $12$.

Calculating the area of a rhombus using the formula of base x Height:
Base $=4$

Height $=2$

Question:

What is the area of the rhombus?

Explanation

The area of the rhombus $=4\times2=8$

Answer

The area of the rhombus is then $8$.

Calculating the area of a rhombus using the formula of base x Height:

Base $=7$
Height $=3$

Question:

What is the area of the rhombus?

Explanation
The area of the rhombus $=7\times3=21$

Answer: $21$

Question:

Find the value of $X=?$

Explanation:

To solve the exercise, we must use the formula for the area of a rhombus, and solve it "backwards".

We will input the data we know into the formula:

$\frac{5\times X}{2}=40$

We'll start by removing the denominator, multiplying the equation by $2$.

$80=X\times5$

Now, we need to isolate $X$. To do this, we divide the equation by $5$.

$X=16$

Answer

The area of the rhombus is then $X=16$.

Question:

What is the area of a rhombus?

Explanation:

To find the solution, we'll need to use another tool to find the missing diagonal.

We know that the diagonals of a rhombus are perpendicular to each other and divide the rhombus into four right triangles.

Therefore, we can examine our triangle and use the Pythagorean theorem to find the missing side.

$A^2+B^2=C^2$

We'll plug in the data we know into the Pythagorean theorem:

$3^2+X^2=5^2$

$9+X^2=25$

Now we'll construct the exercise according to the rules, to solve for $X$.

$25-9=X^2$

$X^2=16$

Now we'll apply the square root to the equation to eliminate the powers

$X=4$

We haven't reached the solution to the exercise yet!

Now that we have found the missing side, we can use the formula for the area of a rhombus to find its area.

(Diagonal A $\times$ Diagonal B): 2

It's important to remember that the data we have is not for the full diagonals, but only from the intersection point of the diagonals to the vertices.

We know that the diagonals of a rhombus also bisect each other, therefore:

Diagonal $1=3\times2=6$

Diagonal $2=4\times2=8$

Now all that's left is to plug the numbers into the formula and solve:

$\left(6\times8\right):2$

$\frac{48}{2}$

$24$

Answer

the area is equal to $24$

Given that the rhombus in the graphic has an area of $24^{}\operatorname{cm}^2$

Task:

What is the value of $X$?

Rhombus area formula

$\frac{diagonal1\times diagonal2}{2}$

Let's plug the data into the formula.

$\frac{8\times X}{2}=24$

$4X=24$ Now we divide by $4$

$X=6$

Finding the diagonals from the formula:

Given the drawing of the rhombus, the area is

$65^{}\operatorname{cm}^2$

Task:

What is the length of the main diagonal in the rhombus?

Solution:

The formula for the area of a rhombus.

(Diagonal times Diagonal)/$2$

$A=\frac{Diagonal1\times Diagonal2}{2}$

When the area is $65$ cm²

We will denote a, which is the main and longer diagonal among the two.

$65=\frac{6.5\times Diagonal2}{2}$ we multiply both sides by $2$

$65\times2=6.5\times Diagonal2$ Now we divide by $6.5$

$Diagonal2=\frac{65\times2}{6.5}=\frac{130}{6.5}=20$

Answer

The length of the main diagonal in the rhombus $=20$

Homework:

What is the area of the rhombus?

Solution:

A rhombus whose pair of adjacent angles equals $90^o$ degrees is a square,

Remember that the area of a square can be calculated using the formula:

Side $\times$ Side = square

We will put the numbers in the formula and solve

$7\times7=49$

Answer:

The area of the rhombus or the square $=49$

Given the drawing of the rhombus, we have the following data as shown $3$ and $5$

Task:

What is the area of the rhombus?

Solution:

To find the solution, we will need to use another tool to get to the missing diagonal.

We know that the diagonals of the rhombus are perpendicular to each other and divide the rhombus into four right triangles.

Therefore, we can look at our triangle and use the Pythagorean theorem to find the missing side.

$A^2+B^2=C^2$

We will present the data we know in the Pythagorean theorem:

$3^2+X^2=5^2$

$9+X^2=25$

Now we will move the numbers to "isolate" the $X$.

$25-9 = X^2$

$X^2 = 16$

We apply the square root to the equation to get rid of the power.

$X=4$

This is still not the solution to the exercise!

Now that we have found the missing side, we can use the formula for the area of a rhombus to find its area.

(Diagonal 1 $\times$ Diagonal 2)

It is important to remember that the data we have is not for all the diagonals, but only from the perspective of the diagonals to the vertices.

We know that the diagonals of the rhombus also intersect each other, therefore

Diagonal 1 $=3\times2=6$

Diagonal 2 $=4\times2=8$

Now we just have to put it in the formula and solve:

$\frac{(6\times8)}{2}$

$\frac{48}{2}$

$24$

Answer:

The area is equal to $24$

Given the drawing of the rhombus

Question:

What is the length of the secondary diagonal?

Solution:

Calculate the area of the rhombus

Area of the rhombus = side X Height

$6\cdot8=48$

It's important to remember that there is another formula to calculate the area of a rhombus

Area of the rhombus = (Diagonal 1 $\times$ Diagonal 2) / $2$

$Area=\frac{d2\times d1}{2}$ we multiply by $\times 2$ (where $d1$ and $d2$ are the diagonals)

$2Area=d2\times d1$; We divide both sides by $d1$

$\frac{2Area}{d1}=d2$

We will introduce the area and the first diagonal into the formula and solve:

$d2=\frac{2Area}{d1}=\frac{2\times48}{6}=\frac{96}{6}=16$

Answer:

The correct answer is $16$ cm.

Topic: Calculating the area of a rhombus using the ratio

Given the rhombus in the figure:

Given that the ratio between the length of the major diagonal and the minor diagonal is $9:2$

Task:

Find the area of the rhombus

Solution:

From the data that the ratio between the diagonals:

Diagonal1 / Diagonal2 $=\frac{9}{2}$

In the given figure for diagonal 2 (the shorter one) we will establish its length in the formula we find.

(Multiplying diagonally) $\frac{9}{2}=\frac{Diagonal1}{4}$

$36=4\times9=2\times Diagonal1$ We will divide by /$:2$

$18=\frac{36}{2}=Diagonal 1$

Let's calculate the area of the rhombus:

$A=\frac{Diagonal 1\times Diagonal 2}{2}=\frac{4\times18}{2}=\frac{72}{2}=36$

Answer:

The answer is $36$ cm²

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• Attendance in math classes at school - essential! Self-study cannot replace learning the material from your teachers. If you have missed a class, make sure to catch up on the taught topic as quickly as possible.
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• Participation in class: essential! The more you participate in class, the better. Why? Because participation keeps you alert and involved in the learning process. Suggest solutions to the questions the teacher raises and ask relevant questions about the study topics.
• Private math class: welcome! No matter what your level is, your grades, and the confidence you have in yourself. Reinforcement classes strengthen your foundations, advance you both material-wise and mentally, and allow you to reap successes and higher grades.
• Plan your learning days before the test well in advance. Prepare a weekly task board, detailing learning hours and learning topics. That way, you'll always arrive ready for the exam and even be able to plan a "study-free day" the day before the exam. The goal: to arrive at the test relaxed, calm, and at peace.

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