Lesson #2 - Augsburg College

Geometry: A Shapely Approach to Math
ever side you draw on the bottom of the paper. Thus, if you rotate the same triangle, each of the three sides
could be the base, depending on how it’s oriented. Now ask your students what the height of the triangle is. The
height is the distance from the point opposite the base to the base. Thus, the height of a triangle may change
depending on which side is the base. Ask your students to draw three of the same triangle, and then mark the
height when the base is each of the three sides. In the diagrams below, the height is represented by the dashed
line. Notice that the dashed line intersects the base at a right angle. As illustrated by the drawing on the right,
you may need to extend the base with a dashed line in order to construct the height. Just remember that the
length of the base is still the length your original line segment.
Now that we know the base and the height of the triangle, how do we use these two numbers to calculate the
area of the triangle? The formula, as your students may know, is base multiplied by height, divided by two. But
how can we see this in a picture? To give your students a hint, ask them if they can turn the triangle into a shape
that they know the area of, like a rectangle.
There are a couple different ways to turn a triangle into a rectangle. Let’s say we begin with the triangle created
by going from point A to B to C. Then draw the height of the triangle, which is the line from B to D, and
two lines parallel to line BD coming up from points A and C. Finally, connect these two lines by making another
new line EF, which is parallel to line AC. Our original triangle ABC now consists of two triangles: ABD
and CBD.
First let’s look at triangle ABD. We see that this triangle is half of the rectangle AEBD. We know the area of
the rectangle AEBD is base times height, which is AD × EA. And since triangle ABD is half that rectangle, its
area is the length of AD (its base) multiplied by the length of BD (its height) divided by two! Now we can do
the same thing for the rectangle BFDC. In conclusion, we can see that the area of triangle ABC is half the area
of rectangle AEFC, and thus we have shown how to calculate the area of a triangle.
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