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In the previous video we discovered that an equilateral triangle of side two has an area
of the square root of three. There were many questions remaining, among them, to determine
the area of an equilateral triangle of side four.
Because the sides have doubled, we could think that the area has double as well and will
be two square root of three. However, the area seems a lot larger. In fact it is. If
we try to fill the triangle of side four with triangles of side two, we quickly see the
Triforce, that mystical relic that will fulfill your wishes if you can touch it.
Rayos! Wait, what I wanted to say is that in the triangle of side four fit four triangles
of side two. This says that the area of the triangle side four is four times larger than
the triangle of side two; it is four square root of three.
Let’s see what happens with an equilateral triangle of side one. Like we did before,
it is possible to fill the triangle of side two with four triangles of side one. This
says that the area of the triangle of side one is one fourth of the triangle side two.
It is square root of three divided by four.
This solves two of the problems stated, even though I find it intriguing that when the
sides are doubled the area multiplies by four. What will happen with the height?
It is possible to calculate the height with the area and base of the triangle. Look at
the formula. We have that the area equals the height times the base divided by two.
Therefore, four square root of three is equal to four times the height divided by two. Four
divided by two is two. And when both sides are divided by two, we find that the height
is two square root of three. Mmm…
When the same is done to the triangle of side one, we find that the square root of three
divided by four is equal to one times the height divided by two. When both sides are
multiplied by two, we find that the height is the square root of three divided by two.
Therefore, we see that when the sides are doubled, the height is doubled.
In fact, we can find the height of any equilateral triangle based on the fact that the height
of the triangle side one is the square root of three divided by two. For example, if the
triangle was of base three, I would know that the height would be three square root of three
divided by two. If the base was six, the height would be three square root of and like that
successively.
When the triangle is enlarged by a factor k that can be three, six or any other number,
we see that its sides and height are enlarged by such factor. And when the area is calculated,
we see that factor multiplied by itself. That is why when the sides are doubled; the area
is multiplied by four.
Then, when the sides are multiplied by three, the area augments nine times and when the
sides are multiplied by four, the area augments sixteen times.