# Area of a right triangle

🏆Practice triangle area

## Formula to find the area of a right triangle

The area of a right triangle is an important subtopic that is repeated over and over again in exercises that include any right triangle.

It is calculated by multiplying the two sides that form the right angle (called legs) and dividing the result by 2.

## Test yourself on triangle area!

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

## Exercise with explanation

For example:

If we have a right triangle whose legs measure $5~cm$ and $6~cm$ and we are asked to find its area, we should multiply $5$ by $6$, giving us a result of 30 and then divide the product by $2$.

That is, the area of the given triangle is $15~cm^2$

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## Exercises to calculate the area of a right triangle

### Exercise 1

Homework:

In front of you is a right triangle, calculate its area.

Solution:

Calculate the area of the triangle using the formula for calculating the area of a right triangle.

$\frac{leg\times leg}{2}$

$\frac{AB\cdot BC}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24$

The answer is $24~cm²$.

### Exercise 2

Homework:

Given the right triangle $\triangle ADB$

The perimeter of the triangle is equal to $30\operatorname{cm}$.

Given:

$AB=15$

$AC=13$

$DC=5$

$CB=4$

Homework:

Calculate the area of the triangle$\triangle~ABC$

Solution:

Given the perimeter of the triangle $Δ~ADC$ equal to $30\operatorname{cm}$.

From here we can calculate $AD$.

$AD+DC+AD=PerimeterΔ~ADC$

$AD+5+13=30$

$AD+18=30$ /$-18$

$AD=12$

Now we can calculate the area of the triangle $Δ~ABC$

Pay attention: we are talking about an obtuse triangle therefore its height is $AD$.

We use the formula to calculate the area of the triangle:

$\frac{height\times side}{2}=$

$\frac{AD\cdot BC}{2}=\frac{12\cdot4}{2}=\frac{48}{2}=24$

The area of the triangle $ΔABC$ is equal to $24~cm²$.

Do you know what the answer is?

### Exercise 3

Homework:

The area of the triangle is equal to $38~cm²$, $AC=8$

Find the measure of the leg $BC$

Solution:

We will calculate the length of $BC$ using the formula for calculating the area of the right triangle:

$\frac{leg\times leg}{2}$

$\frac{AC\cdot BC}{2}=\frac{8\cdot BC}{2}=38$

We multiply the equation by the common denominator

/ $\times2$

Then we divide the equation by the coefficient of $BC$

8\timesBC=76 /$:8$

$BC=9.5$

The length of the leg $BC$ is equal to $9.5$ centimeters.

### Exercise 4

In front of you, there is a right triangle $Δ~ABC$.

Given that $BC=6$. The length of the leg $AB$ is greater by $33\frac{1}{3}\%$ than the length of $BD$.

The area of the triangle $\triangle~ADC$ is greater by $25\%$ than the area of the triangle $\triangle~ABD$.

What is the area of the triangle $\triangle~ABC$?

Solution:

To find the measure of the leg $AB$ we will use the data that its length is greater by $33.33$ than the length of $BD$.

$AB=1.33333\cdot BD$

$(\frac{100}{100}+\frac{33.33}{100}=\frac{133.33}{100}=1.333)$

$AB=1.333\cdot6=8$

Now we will calculate the area of the triangle $ΔABD$.

$A~Δ\text{ABD}=\frac{AB\cdot BD}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24$

$24~cm²$.

### Exercise 5

Homework:

Which data in the graph is incorrect?

For the area of the triangle to be $24~cm²$, what is the data that should be in place of the error?

Solution:

Explanation: area of the right triangle.

$AΔEDF=\frac{ED\cdot EF}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24$

According to the formula:

$\frac{leg\times leg}{2}$

If the area of the triangle can also be calculated from the formula of:

$\frac{side\times side~height}{2}$

$\frac{EG\times10}{2}=24$ /$\times2$

$10EG=48$ /$:10$

$EG=4.8$

The incorrect data is $EG$.

The length of $EG$ should be $4.8\operatorname{cm}$.

If you are interested in learning more about other triangle topics, you can enter one of the following articles:

In the blog of Tutorela you will find a variety of articles about mathematics.

Do you think you will be able to solve it?

## examples with solutions for area of a right triangle

### Exercise #1

Calculate the area of the right triangle below:

### Step-by-Step Solution

As we see that AB is perpendicular to BC and forms a 90-degree angle

It can be argued that AB is the height of the triangle.

Then we can calculate the area as follows:

$\frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24$

24 cm²

### Exercise #2

Calculate the area of the triangle ABC using the data in the figure.

### Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

(the side * the height that descends to the side) /2

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

$\frac{CB\times AD}{2}$

$\frac{8\times9}{2}=\frac{72}{2}=36$

36 cm²

### Exercise #3

Calculate the area of the following triangle:

### Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

$\frac{BC\times AE}{2}$

Now we replace the existing data:

$\frac{4\times5}{2}=\frac{20}{2}=10$

10

### Exercise #4

Calculate the area of the triangle below, if possible.

### Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Cannot be calculated

### Exercise #5

What is the area of the given triangle?

### Step-by-Step Solution

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

The height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

$\frac{6\times5}{2}=\frac{30}{2}=15$