The area of a triangle

​ ​Well, here you are munching grass. Which goat has the most grass in its pasture? In other words, which of these triangle has the largest area or surface? If you draw these three triangles on a sheet of paper and cut them out, you can weigh them on a scale. The heaviest triangle also has the largest area assuming you have used the same paper for all of them of course.

This is a little awkward perhaps and sometimes you need to know exactly what is the area not just which figure is the largest. In this case, you can not cut and weigh. You have to calculate. This is when it's useful to learn how to calculate the area of a rectangle. The area of rectangle is its length times its width, of course.

If you do the same with this triangle, multiply its base by its height, you get this area. You can then split it apart like this: a diagonal from corner to corner splits the rectangle into two equal parts. These two parts are exactly the same. Therefore, the triangle's area is half the rectangle's area, or the base times height divided by two. This works for all triangles.

But there are a few things that can complicate this problem. ​ ​For this particular triangle, it's easy to see what the height is. Since it's a right angle, this leg is perpendicular to the base. Therefore, the leg's length is also the triangle's height. But what if we do this? What is the triangle's height now?

Is this still the height? No, it's not. The height must make the right angle with the base. The area of this triangle just like every other triangle, is the base times height, divided by two. With the right triangle, it was easy to see how it works.

If it's hard to believe that it works for this triangle as well, think of it this way: split the triangle straight down from its tallest point. You now have two right-angled triangles and each of them is half of the base times height. If you just multiply the base by the height, you get an area that is exactly two times larger than the area of the triangle. So, the triangle's area is equal to the base times height divided by two. ​ ​What about this triangle, how do you calculate the area of this one? Here, you can't draw a vertical line from the base to its top.

There are two ways to solve this. You could measure the height from an imaginary line that continues the base like this, then take the base times height divided by two. Or, you could simply do it like this. Now it's easy to find the height and take the base times height divided by two. The area is the same.

If you remember this, you can always calculate the area of a triangle. The base times height divided by two.