The formulae for the area of a triangle. (Part 3 – Scalene.)

In the previous blogs we have seen that the area for any right-triangle and any isosceles triangle (to include the special equilateral triangle) has given up the formula area = bh/2. We should now move on to the next type of triangle, the scalene to discover the area formula for this version of the three sided polygon. In fact there are two distinct forms of scalene triangle, the acute scalene and the obtuse scalene (see below.)

acobt

In the acute scalene all angles are less than 90˚ and in the obtuse one of the angles is greater than 90˚. We are still trying to find a general formula for each triangle that will help us to calculate its area. So far with the right-triangle and the isosceles the formula for their areas has been the same (bh/2). Students might find that the use of a geoboard could help on this leg of the journey.

Start by getting them to make an acute scalene triangle on a geoboard. 

scalene1

As this side of the geoboard has pins perpendicular to each other, this triangle can be wrapped in a rectangle with a base the same length as the triangle and a height the same length as the vertical height of the triangle. 

scalene2                                     scalene21

Because we have a rectangle surrounding our triangle which has one of its vertices lying on a length of the rectangle we can use this to drop a perpendicular line down to the base length at the same time creating two rectangles within the greater rectangle.

scalene3                                      scalene32

If we look at the last diagram it is clear that each of the two smaller rectangles each contain two identical right-triangles. At this point we can make a link to algebra to prove the area formula for the acute scalene triangle. Start by labelling the identical triangles within the greater rectangle, say A for the small right-triangle and B for the larger one.

algebra

 

The area for the greater rectangle could now be defined as 2A + 2B. This can be factorised.

algebra

On inspection it is clear that the area of the acute scalene triangle we started with has an area of A + B which is exactly half of the area of the greater rectangle. It is therefore true yet again that the formula for the area of this triangle is exactly the same as the formulae for the right-triangle and isosceles, half of the rectangle that surrounds it, bh/2 where b is the base length and h is the vertical height of the triangle. But what about the other scalene triangle? The obtuse scalene. We cannot take it for granted that the same formulae for the area of the other triangles will fit this one too. The geoboard can help to discover the formula however. We can start by asking students to form an obtuse (one angle greater than 90˚) triangle on the geoboard. This one may not be as straight forward as we would prefer it to be. However the triangle can again be inscribed in a rectangle.

scalobtuse1                 scalobtuse2

Unfortunately the only thing that the two diagrams have in common are the vertical heights. It can be seen that the base length of the rectangle is slightly bigger than that of the triangle. With this triangle we need to use information that we know to be true. Namely that the area of the rectangle is found by multiplying the lengths of its base and height. Firstly, however we should label the dimensions in the rectangle (see below.)

rec1

The area for the rectangle is h(a + b) or ah + bh. The rectangle is made up of three triangles, two right-triangles (and we know how to calculate their areas) plus the triangle whose area we are trying to determine. We should start by calculating the area of each of the right-triangles. The small triangle at the left has an area ah/2. The larger of the two right-triangles at the right has an area h(a + b)/2 which can be written ah/2 + ab/2. If we add these two area and subtract the result from the area of the rectangle we will be left with the area of the obtuse scalene triangle. Now if this seems a bit messy look at what is happening. The following diagram in three parts sets out to explain the reasoning for the watertight proof of the area for an obtuse scalene triangle.

proof5

Part 1

This explains the area of the rectangle in terms of the dimensions given. It also states that the area also comprises the three areas of the different triangle that together make the rectangle.

Part 2

Shows the dimensions of the rectangle and triangles and defines the areas of triangles A and B.

Part 3

When the algebraic areas of the sum of triangles A and B are subtracted from the area of the rectangle (A+B+C) we are left with area C which is equal the length b multiplied by length h then all divided by 2. As you will notice length b happens to be the base of the obtuse scalene triangle and h is its vertical height. We have now successfully deduced that the area for any triangle can be calculated by multiplying base length by vertical height followed by division by 2.

 

 

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