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<a id="backhome" href='/' >
Greater Skies
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<p>
<div id="related">
<a href="st-es.html">Versión en castellano</a>.
</p>
<p>
See also
</p><ul>
<li>
<a href="../six-times-nine/stn-en.html">Why is six times nine the same as nine times six?</a>
</li>
<li>
<a href="../size-paral/sp-en.html">What is the size of a parallelogram?</a>
</li>
</ul>
<p></div>
</p>
<div id="outline-container-1" class="outline-2">
<h2 id="sec-1">What is the size of a triangle? </h2>
<div class="outline-text-2" id="text-1">
<p>Say you want to know how big this triangle is:
</p>
<p>
<img src="img/triangle-1.png" Class="clean-center" alt="img/triangle-1.png" />
</p>
<p>
Or whether it is bigger or smaller than this other one:
</p>
<p>
<img src="img/triangle-2.png" class="clean-center" alt="img/triangle-2.png" />
</p>
<p>
What does size mean when you are talking about a triangle? It
can be how much paint you need to color it, or how many kilos
of seed you need to plant on it to cover it with grass, or how
many rectangular cookies you can fit on it. All these measures
are quite arbitrary, but they can all make sense on certain
occasions.
</p>
<p>
The arbitrary measure we tend to stick with is: how many
squares can you fit on it? You pick a square of a given side,
say 1 m or 1 km, try to figure out how many of them will fit in
the triangle, and call it the <b>area of the triangle</b>. Filling
up a triangle with squares is not easy, as squares have the
annoying property not to fit perfectly inside a triangle.
</p>
<p>
Squares do not fit triangles well, but they do a pretty good
job filling up rectangles, so we might think about adding a
rectangle to our triangle, like this:
</p>
<p>
<img src="img/triangle-3.png" class="clean-center" alt="img/triangle-3.png" />
</p>
<p>
This is the minimum rectangle that encloses our triangle. Now
we have a triangle and a rectangle that are clearly related: if
we make the triangle bigger, the rectangle enclosing it will be
also bigger.
</p>
<p>
Shouldn't there be a relationship between the size of the
rectangle and the size of the triangle? Because we know how to
measure a rectangle: we <a href="six-times-nine.html">multiply its sides</a>, in this case
<img src="ltxpng/st-en_c32770d5b8b96729ed7d6ac209b867c8c92df510.png"/>. If we could figure out the relationship between
their sizes, we would know the size of the triangle.
</p>
<p>
Take a look at this other image:
</p>
<p>
<img src="img/triangle-4.png" class="clean-center" alt="img/triangle-4.png" />
</p>
<p>
Do you see it already? If not, try this other one:
</p>
<p>
<img src="img/triangle-5.png" class="clean-center" alt="img/triangle-5.png" />
</p>
<p>
We have divided the rectangle in two smaller rectangles, the
green and the blue, and at the same time we have divided our
triangle in two smaller triangles, also green and blue. Each
small triangle is exactly half its small rectangle.
</p>
<p>
In other words: the size of our triangle is exactly half the
size of the rectangle. So, if you want to measure a triangle,
you just multiply its base <img src="ltxpng/st-en_f46f0eb142f1e88fce09b363508e32381833ebbc.png"/> times its height <img src="ltxpng/st-en_627f79684ded3e16a85d44bf8bbd232ea90c2b77.png"/>, and then
divide by two.
</p>
<p>
But wait: this argument surely does not work for the other
triangle we had up there, the one we wanted to compare with our
original triangle:
</p>
<p>
<img src="img/triangle-2.png" class="clean-center" alt="img/triangle-2.png" />
</p>
<p>
True, the argument doesn't hold, but the way to measure it
turns out to be the same: we multiply <img src="ltxpng/st-en_c32770d5b8b96729ed7d6ac209b867c8c92df510.png"/>, and divide
by two. But you have to choose your <img src="ltxpng/st-en_f46f0eb142f1e88fce09b363508e32381833ebbc.png"/> and <img src="ltxpng/st-en_627f79684ded3e16a85d44bf8bbd232ea90c2b77.png"/> wisely, like
this:
</p>
<p>
<img src="img/triangle-6.png" class="clean-center" alt="img/triangle-6.png" />
</p>
<p>
So, if this is true, you can move the upper vertex of a
triangle horizontally as much as you want, and as long as <img src="ltxpng/st-en_627f79684ded3e16a85d44bf8bbd232ea90c2b77.png"/>
and <img src="ltxpng/st-en_f46f0eb142f1e88fce09b363508e32381833ebbc.png"/> remain constant its area will not change.
</p>
<p>
But is it true? It is a great exercise, so maybe you can think
about it. <a href="../size-paral/sp-en.html">Hint</a>.
</p></div>
</div>
<address class="signature">
<a class="author" href="http://juanreyero.com">Juan Reyero</a>
<span class="date">
Barcelona,
2008-04-15</span>
</address>
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