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.

A - area of a new right triangle

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Test yourself on triangle area!

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Complete the sentence:

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

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Right triangle

Exercise with explanation

For example:

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

That is, the area of the given triangle is 15 cm2 15~cm^2

area of the given triangle is 15


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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.

a right triangle, calculate its area

Solution:

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

leg×leg2 \frac{leg\times leg}{2}

ABBC2=862=482=24 \frac{AB\cdot BC}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24

Answer:

The answer is 24 cm2 24~cm² .


Exercise 2

Homework:

Given the right triangle ADB \triangle ADB

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

Given:

AB=15 AB=15

AC=13 AC=13

DC=5 DC=5

CB=4 CB=4

Homework:

Calculate the area of the triangle ABC \triangle~ABC

Given the right triangle ADB

Solution:

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

From here we can calculate AD AD .

AD+DC+AD=PerimeterΔ ADC AD+DC+AD=PerimeterΔ~ADC

AD+5+13=30 AD+5+13=30

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

AD=12 AD=12

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

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

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

height×side2= \frac{height\times side}{2}=

ADBC2=1242=482=24 \frac{AD\cdot BC}{2}=\frac{12\cdot4}{2}=\frac{48}{2}=24

Answer:

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


Do you know what the answer is?

Exercise 3

Homework:

Given the right triangle Δ ABC Δ~ABC

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

Find the measure of the leg BC BC

A=38 cm²

Solution:

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

leg×leg2 \frac{leg\times leg}{2}

ACBC2=8BC2=38 \frac{AC\cdot BC}{2}=\frac{8\cdot BC}{2}=38

We multiply the equation by the common denominator

/ ×2 \times2

Then we divide the equation by the coefficient of BC BC

8\timesBC=76 /:8 :8

BC=9.5 BC=9.5

Answer:

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


Exercise 4

Exercise 4 In front of you is a right triangle ABC

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

Given that BC=6 BC=6 . The length of the leg AB AB is greater by 3313% 33\frac{1}{3}\% than the length of BD BD.

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

Task:

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

Solution:

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

AB=1.33333BD AB=1.33333\cdot BD

(100100+33.33100=133.33100=1.333)(\frac{100}{100}+\frac{33.33}{100}=\frac{133.33}{100}=1.333)

AB=1.3336=8 AB=1.333\cdot6=8

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

A ΔABD=ABBD2=862=482=24 A~Δ\text{ABD}=\frac{AB\cdot BD}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24

Answer:

24 cm2 24~cm² .


Check your understanding

Exercise 5

the area of the triangle is 24 cm²

Homework:

Which data in the graph is incorrect?

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

Solution:

Explanation: area of the right triangle.

AΔEDF=EDEF2=862=482=24 AΔEDF=\frac{ED\cdot EF}{2}=\frac{8\cdot6}{2}=\frac{48}{2}=24

According to the formula:

leg×leg2 \frac{leg\times leg}{2}

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

side×side height2 \frac{side\times side~height}{2}

EG×102=24 \frac{EG\times10}{2}=24 /×2 \times2

10EG=48 10EG=48 /:10 :10

EG=4.8 EG=4.8

Answer:

The incorrect data is EG EG .

The length of EG EG should be 4.8cm 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:

101010666888AAACCCBBB

Video Solution

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:

AB×BC2=8×62=482=24 \frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24

Answer

24 cm²

Exercise #2

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

121212888999AAABBBCCCDDD

Video Solution

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:

CB×AD2 \frac{CB\times AD}{2}

8×92=722=36 \frac{8\times9}{2}=\frac{72}{2}=36

Answer

36 cm²

Exercise #3

Calculate the area of the following triangle:

444555AAABBBCCCEEE

Video Solution

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:

BC×AE2 \frac{BC\times AE}{2}

Now we replace the existing data:

4×52=202=10 \frac{4\times5}{2}=\frac{20}{2}=10

Answer

10

Exercise #4

Calculate the area of the triangle below, if possible.

8.58.58.5777

Video Solution

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.

Answer

Cannot be calculated

Exercise #5

What is the area of the given triangle?

555999666

Video Solution

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:

A1- How to find the area of a triangleThe 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:

6×52=302=15 \frac{6\times5}{2}=\frac{30}{2}=15

Answer

15

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