Area of a Scalene Triangle

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Area of a scalene triangle

Formula to calculate the area of a scalene triangle:

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

121212888999AAABBBCCCDDD

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Area of the scalene triangle

It's very simple to calculate the area of a scalene triangle if we remember the formula and strictly follow the steps. Don't worry, we're here to teach you exactly what to pay attention to—we won't leave you adrift!
First of all, let's look at the formula you need to remember in order to calculate the area of the scalene triangle:


Multiply the height by the base (the side corresponding to that height) and divide by 22.

Pay attention:

Make sure to place in the formula the corresponding height and side. That is, if a certain height and a side that does not form a right angle of 90o 90^o degrees with the used height is placed in the formula, it will be wrong.


Let's see it in an exercise

Given the triangle ABCABC
Given that:
DB=6DB=6 Height
AC=7AC = 7
What is the area of the triangle?

Solution:
We will see that the given side ACAC actually forms, with the height, an angle of 90o 90^o  degrees.
After verifying the data, we will go to the formula and place there:
6×72=21\frac{6\times7}{2}=21

The area of the triangle ABCABC is 21cm2 21\operatorname{cm}^2


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Now we will calculate the area of a right triangle

Given the right triangle EFGEFG
Given that:
angle EFG=90EFG = 90
EF=5EF=5
FG=6FG=6
Calculate the area of the triangle.

Solution:
Let's remember that the key to calculating the area of any triangle is to multiply the height
by the corresponding side and then divide that product by22
In a right triangle, we actually already have the height!
We don't need to calculate another height and, in fact, we can afford to use the given height along with the side that forms the 90o 90^o degree angle.

In our exercise: The side is EFEF or FGFG

What conclusion do we reach?
The conclusion is that the formula to calculate the area of a right triangle is the product of the two legs divided by 22Let's put it in the formula and we will get:

6×52=15 \frac{6\times5}{2}=15
The area of the triangle EFGEFG is 15cm2 15cm^2


Now let's move on to calculating the area of an obtuse triangle

Calculating the area of an obtuse triangle is a bit more complicated, but I assure you that once you understand the basic principle, you will be able to calculate the area of an obtuse triangle even in your sleep...
In certain cases, in an obtuse triangle, we will be given a height that is outside the triangle.
As in the following illustration:

In this illustration, the height AGAG has been drawn outside of the triangle. In reality, if we were to extend the side CBCB (marked in green), it would form a right angle with the height.
How is the area of an obtuse triangle calculated?

Remember the following guidelines and you will do well:

  • In calculating the area of the obtuse triangle, we refer to the actual side length of the triangle and not to its dotted extension.
  • In calculating the area of the obtuse triangle, we refer to the given height (even if it is outside the triangle) and look for the corresponding side, which together with it forms a 90o 90^o degree angle when extended outside the triangle.

Now let's solve an exercise so you can understand it more easily:

Given the triangle ABC \triangle ABC
Given that:
BD=2BD= 2 Height of the triangle
AD=5AD= 5
CD=12CD= 12

What is the area of the triangle?

Solution:
We observe that the length of the side DB=2DB = 2
and the corresponding side that forms with it a 90o 90^o degree angle (the dotted part outside the triangle) is CACA
If we go back to the first point we needed to remember - we will understand that, to calculate the area, we must only take into account the length of ACAC without its dotted extension.
Therefore, we will see it as 125=7 12-5=7
AC=7AC=7
And now we can safely place the data, according to the basic formula:
7×22=7\frac{7\times2}{2}=7
The area of the triangle ABCABC is 7cm2 7cm^2


Examples and exercises with solutions for calculating the area of a scalene triangle

examples.example_title

Find the area of the triangle (note that this is not always possible)

8.58.58.5777

examples.explanation_title

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.

examples.solution_title

Cannot be calculated

examples.example_title

Which of the following triangles have the same area?

101010121212555131313555888121212666666FFFEEEGGGCCCBBBAAAKKKJJJIIIDDDLLLHHH

examples.explanation_title

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

It can be seen that after the calculation, the areas of the similar triangles are ABC and EFG

examples.solution_title

EFG, ABC

examples.example_title

Given the triangle PRS

The length of side SR is 4 cm

The area of the triangle PSR is 30 cm²

Calculate the height PQ

S=30S=30S=30444PPPRRRSSSQQQ

examples.explanation_title

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

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

LadoAltura2=Aˊrea del triangulo \frac{Lado\cdot\text{Altura}}{2}=Área~del~triangulo

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

examples.solution_title

15 cm

examples.example_title

Find X using the data from the figure:

S=20S=20S=20555XXXAAABBBCCC

examples.explanation_title

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

examples.solution_title

8

examples.example_title

The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

It is known that the ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC

AAABBBCCCDDDEEE

examples.explanation_title

To calculate the ratio between the sides we will use the existing figure:

AAEDAABCE=13 \frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

AADE=h×DE2 A_{ADE}=\frac{h\times DE}{2}

We know that the area of the parallelogram is equal to:

AABCD=h×EC A_{ABCD}=h\times EC

We replace the data in the formula given by the ratio between the areas:

12h×DEh×EC=13 \frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}

We solve by cross multiplying and obtain the formula:

h×EC=3(12h×DE) h\times EC=3(\frac{1}{2}h\times DE)

We open the parentheses accordingly

h×EC=1.5h×DE h\times EC=1.5h\times DE

We divide both sides by h

EC=1.5h×DEh EC=\frac{1.5h\times DE}{h}

We simplify to h

EC=1.5DE EC=1.5DE

Therefore, the ratio betweenECDE=11.5 \frac{EC}{DE}=\frac{1}{1.5}

examples.solution_title

1:1.5 1:1.5

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