The area of triangle

 We measure an area with squares; therefore we calculate the area of a triangle in the same way.

The area of a triangle is calculated by seeing how many squares can fit inside the triangle.


The triangle above consists of 30 unit squares and 12 half squares. Since 12 half squares equal 6 whole squares, we can say that the square above is 36 unit squared.



Not all triangles may be drawn in such a neat manner, so some squares may not equal half squares, and also constantly drawing squares within a triangle to find the area is troublesome and takes time.


( Please pay attention triangles are not the same )

We must find the solution in a more practical manner…

Fundamental mathematics for area:

If you multiply the foundation/base with the length; it is as if you have
put the foundation on top of one another as much as the length.


You can think about this as stacking squares on top of one another starting from the
base.

Now let us divide the area into two equal parts.



If you divide a rectangle (a square is also a rectangle) starting from the edges you will have equal parts, thus equal areas.



Try it !
Draw a rectangle on a piece of paper, cut it, now see that two pieces are the same!




What does dividing a rectangle from edge to edge do for me?

We have created two triangles with an equal area. In other words, the triangle is
Half of the rectangle, we can use this to calculate the area of the triangle.


Now we have learnt the fundamentals, we can start to calculate the area of triangles.

Areas in acute angled triangles:





When we multiply the base with the length, we find the area of the rectangle within the red area,
whereas we only want to find out the area of the triangle.



The area of the rectangle consists of two stars and two circles, whereas the
triangle consists of one star and one circle; so the area of the triangle is half of the area of the rectangle.

Areas in right-angled triangle: 


The area of the triangle is half of the rectangle.

Obtuse angled triangle;

For example in this triangle;


If we accept that when we multiply the base with the length, I am creating an equal area at the top the same as the base;



It is hard to see whether or not the area marked as ??? covers the shaded area  completely or not.

Let us find a different method:


Let’s imagine that there is an Extension next to the base.



( extension + base) * Height gives us the area of the triangle .

Now we are going to do some mathematics, you may get confused, but if you try, 
I am sure you will understand .

Our goal is to find the area of ;





Base . Height must be  ; 



Our goal is to find the area of ;



\( \displaystyle \frac{Base*Height}{2} \) = 

So ...


\( \displaystyle\text{The are of triangle = }\frac{Base*Height}{2} \)

Area of triangle practice


Now that we have learnt how to calculate the length of a triangle, let’s practice.

When finding the area of a triangle the base value must be calculated precisely. 


Areas in acute angled triangles.





\( \displaystyle\text{The are of triangle = }\frac{6cm*8cm}{2}=24cm² \)


\( \displaystyle\text{The are of triangle = }\frac{12cm*9cm}{2}=54cm² \)


\( \displaystyle\text{The are of triangle = }\frac{14cm*8cm}{2}=56cm² \)

Find height of triangle when given area


Let us find the length of the place marked as ?? .




\( \displaystyle\text{The are of triangle = }\frac{9cm*10cm}{2}=45cm² \)

I can also find the same area, when the side base and base length is 15 cm.

\( \displaystyle\text{45cm²= }\frac{15cm*??}{2} \)

\( \displaystyle 15cm.??=90cm² \)

\( \displaystyle ??=6cm \)

Areas in Right-angled triangles


The base is 12 cm , the height is 6 cm in the given triangle .

\( \displaystyle\text{The are of triangle = }\frac{6cm*12cm}{2}=36cm² \)


\( \displaystyle\text{The are of triangle = }\frac{20cm*12cm}{2}=120cm² \)


Areas in obtuse angled triangles:



\( \displaystyle\text{The are of triangle = }\frac{8cm*9cm}{2}=36cm² \)




\( \displaystyle\text{The are of triangle = }\frac{18cm*5cm}{2}=45cm² \)




\( \displaystyle\text{The are of triangle = }\frac{14cm*6cm}{2}=42cm² \)



\( \displaystyle\text{The are of triangle = }\frac{10cm*9cm}{2}=45cm² \)


You must find the base and the height of the base.
All other lengths are insignificant when calculating the area of a triangle.

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