Area of a right triangle - Examples, Exercises and Solutions

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.

Practice 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

Exercise #1

What is the area of the triangle in the drawing?

5557778.68.68.6

Video Solution

Step-by-Step Solution

First, we will identify the data points we need to be able to find the area of the triangle.

the formula for the area of the triangle: height*opposite side / 2

Since it is a right triangle, we know that the straight sides are actually also the heights between each other, that is, the side that measures 5 and the side that measures 7.

We multiply the legs and divide by 2

5×72=352=17.5 \frac{5\times7}{2}=\frac{35}{2}=17.5

Answer

17.5

Exercise #2

Calculate X using the data in the figure below.

S=20S=20S=20555XXXAAABBBCCC

Video Solution

Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height descending from the side) /2

We place the data we have into the formula to find X:

20=AB×AC2 20=\frac{AB\times AC}{2}

20=x×52 20=\frac{x\times5}{2}

Multiply by 2 to get rid of the fraction:

5x=40 5x=40

Divide both sections by 5:

5x5=405 \frac{5x}{5}=\frac{40}{5}

x=8 x=8

Answer

8

Exercise #3

The triangle ABC is given below.
AC = 10 cm

AD = 3 cm

BC = 11.6 cm
What is the area of the triangle?

11.611.611.6101010333AAABBBCCCDDD

Video Solution

Step-by-Step Solution

The triangle we are looking at is the large triangle - ABC

The triangle is formed by three sides AB, BC, and CA.

Now let's remember what we need for the calculation of a triangular area:

(side x the height that descends from the side)/2

Therefore, the first thing we must find is a suitable height and side.

We are given the side AC, but there is no descending height, so it is not useful to us.

The side AB is not given,

And so we are left with the side BC, which is given.

From the side BC descends the height AD (the two form a 90-degree angle).

It can be argued that BC is also a height, but if we delve deeper it seems that CD can be a height in the triangle ADC,

and BD is a height in the triangle ADB (both are the sides of a right triangle, therefore they are the height and the side).

As we do not know if the triangle is isosceles or not, it is also not possible to know if CD=DB, or what their ratio is, and this theory fails.

Let's remember again the formula for triangular area and replace the data we have in the formula:

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

Now we replace the existing data in this formula:

CB×AD2 \frac{CB\times AD}{2}

11.6×32 \frac{11.6\times3}{2}

34.82=17.4 \frac{34.8}{2}=17.4

Answer

17.4

Exercise #4

Which of the following triangles have the same areas?

101010121212555131313555888121212666666FFFEEEGGGCCCBBBAAAKKKJJJIIIDDDLLLHHH

Video Solution

Step-by-Step Solution

We calculate the area of triangle ABC:

12×52=602=30 \frac{12\times5}{2}=\frac{60}{2}=30

We calculate the area of triangle EFG:

6×102=602=30 \frac{6\times10}{2}=\frac{60}{2}=30

We calculate the area of triangle JIK:

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

Therefore, the triangles that have the same areas are ABC and EFG.

Answer

EFG and ABC

Exercise #5

The area of triangle ABC is 20 cm².

Its height (AD) is 8.

Calculate the length of the side BC.

S=20S=20S=20888AAACCCBBBDDD

Video Solution

Step-by-Step Solution

We can present the data in the formula to calculate the area of the triangle:

S=AD×BC2 S=\frac{AD\times BC}{2}

20=8×BC2 20=\frac{8\times BC}{2}

Cross multiplication:

40=8BC 40=8BC

Divide both sides by 8:

408=8BC8 \frac{40}{8}=\frac{8BC}{8}

BC=5 BC=5

Answer

5 cm

Exercise #1

PRS is a triangle.

The length of side SR is 4 cm.
The area of triangle PSR is 30 cm².

Calculate the height PQ.

S=30S=30S=30444PPPRRRSSSQQQ

Video Solution

Step-by-Step Solution

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

Pay attention: in an obtuse triangle, the height is located outside of the triangle!

SideHeight2=Triangular Area \frac{Side\cdot Height}{2}=\text{Triangular Area}

Double the equation by a common denominator:

4PQ2=30 \frac{4\cdot PQ}{2}=30

2 \cdot2

Divide the equation by the coefficient of PQ PQ .

4PQ=60 4PQ=60 / :4 :4

PQ=15 PQ=15

Answer

15 cm

Exercise #2

Below is the right triangle ABD, which has a perimeter of 36 cm.

AB = 15

AC = 13

DC = 5

CB = 4

Work out the area of the triangle.

151515444555131313BBBCCCDDDAAA

Video Solution

Step-by-Step Solution

According to the data:

BD=4+5=9 BD=4+5=9

Now that we are given the perimeter of triangle ABD we can find the missing side AD:

AD+15+9=36 AD+15+9=36

AD+24=36 AD+24=36

AD=3624=12 AD=36-24=12

Now we can calculate the area of triangle ABD:

AD×BD2=12×92=1082=54 \frac{AD\times BD}{2}=\frac{12\times9}{2}=\frac{108}{2}=54

Answer

54 cm²

Exercise #3

The triangle ABC has a perimeter measuring 42 cm.

AD = 12

AC = 15

AB = 13

Calculate the area of the triangle.

131313151515121212AAABBBCCCDDD

Video Solution

Step-by-Step Solution

Given that the perimeter of triangle ABC is 42.

We will use this data to find side CB:

13+15+CB=42 13+15+CB=42

CB+28=42 CB+28=42

CB=4228=14 CB=42-28=14

Now we can calculate the area of triangle ABC:

AD×BC2=12×142=1682=84 \frac{AD\times BC}{2}=\frac{12\times14}{2}=\frac{168}{2}=84

Answer

84 cm²

Exercise #4

Calculate the area of the

triangle ABC given that its

perimeter equals 26.

666AAABBBCCCEEE97

Video Solution

Step-by-Step Solution

Remember that the perimeter of a triangle is equal to the sum of all sides together,

Now find side BC:

26=9+7+BC 26=9+7+BC

26=16+BC 26=16+BC

We move the 16 to the left section and keep the corresponding sign:

2616=BC 26-16=BC

10=BC 10=BC

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

(the side * the height) /2

That is:

BC×AE2 \frac{BC\times AE}{2}

We replace the existing data:

10×62=602=30 \frac{10\times6}{2}=\frac{60}{2}=30

Answer

30

Exercise #5

Triangle ABC is a right triangle.

The area of the triangle is 6 cm².

Calculate X and the length of the side BC.

S=6S=6S=6444X-1X-1X-1X+1X+1X+1AAACCCBBB

Video Solution

Step-by-Step Solution

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

ACBC2=cateto×cateto2 \frac{AC\cdot BC}{2}=\frac{cateto\times cateto}{2}

And compare the expression with the area of the triangle 6 6

4(X1)2=6 \frac{4\cdot(X-1)}{2}=6

Multiplying the equation by the common denominator means that we multiply by 2 2

4(X1)=12 4(X-1)=12

We distribute the parentheses before the distributive property

4X4=12 4X-4=12 / +4 +4

4X=16 4X=16 / :4 :4

X=4 X=4

We replace X=4 X=4 in the expression BC BC and

find:

BC=X1=41=3 BC=X-1=4-1=3

Answer

X=4, BC=3

Topics learned in later sections

  1. Area
  2. The Sum of the Interior Angles of a Triangle
  3. The sides or edges of a triangle
  4. Triangle Height
  5. Exterior angle of a triangle
  6. Obtuse Triangle
  7. Identification of an Isosceles Triangle
  8. Scalene triangle
  9. Acute triangle
  10. Area of Isosceles Triangles
  11. Area of a Scalene Triangle
  12. Area of Equilateral Triangles
  13. Perimeter
  14. Triangle