Finding the Area of an Acute Triangle

Flying above the city of Polygon, Pennsylvania Caroline the Consultant sees an opportunity to add some flair to the city's skyline. The tops of Polygon's triangular skyscrapers are just so plain. What a waste of primo advertising space! So Caroline the Consultant brings a design plan to the mayor of Polygon. Why not turn these ugly old rooftops into sky-high welcome mats, greeting visitors to the city! To design effective advertisements for each skyscraper, we'll have to measure the size of their rooftops which means finding the area of acute triangles. The tops of both skyscrapers are shaped like acute triangles. Acute triangles are triangles where ALL the angles are smaller than 90 degrees. Another kind of triangle--which has exactly one 90 degree angle--is called a RIGHT triangle. A right triangle looks a bit like a rectangle, if it was cut in half with a diagonal. So how would you calculate the area of a right triangle? Since right triangles are half of a rectangle, we can find their area by multiplying one half the length of the base times the height. Let's see if we can use this information to figure out the area of our acute rooftops. Look closely at the acute triangle on the left: can you see a way to break this up into RIGHT triangles? If we draw a line perpendicular to the base, we can create two separate right triangles. Notice that this line is the height of both right triangles and the acute triangle itself because it is perpendicular to the bases of each. Now to find the area of the acute triangle, all we have to do is add together the areas of these two right triangles. Let's start with the triangle on the left: what numbers should we use for our base and height? The length of the bottom side is 20, so we'll use that for our base. The line we drew perpendicular to that side has a length of 10, so we'll use that for our height. Multiplying, we see this triangle takes up 100 square meters. The triangle on the right has a base of 4 and a height of 10, giving it an area of 20 square meters. If we add the areas of both triangles together, the total area is 120 square meters. Before we move on, let's see if we can find a more general formula to make our work easier in the future. Notice that the base of the whole acute triangle is 24. If we substitute that and the height of 10 into our area formula, we get the same answer! That means that just like right triangles, the area of our acute triangle is ALSO half the area of a rectangle with the same base and height. To understand WHY this works, let's circumscribe a rectangle around our acute triangle. With our height line still drawn, we now have four right triangles. The two right triangles on the left make up two halves of a rectangle and the two triangles on the right form another rectangle. So our original triangle takes up exactly half the space of a rectangle with the same base and height which we can see here as two identical acute triangles. This is why the area of any acute triangle will always be one half the base times the height. Let's keep this in mind, as we calculate the area of the second rooftop. In order to use our area formula, we need to find the base and height of this acute triangle. So what number should we use for our base? This ENTIRE side is our base, so we're going to have to combine 18 and 22 to get the WHOLE side length of 40 meters. We can substitute that into our formula for the base, 'b'. Now which number should we use for the height? When looking for the height, always keep an eye out for the right angle symbol, which indicates that two lines are perpendicular. That makes our height, 'h', 24. Multiplying together we see this rooftop has an area of 480 square meters! While those welcome signs are being built, let's review. Just like with right triangles, to find the area of an acute triangle, we can use the formula, one-half base times height. This works, because every acute triangle is composed of two right triangles which are themselves each one half of a rectangle. Finally, when identifying the base and height of any triangle, make sure they are perpendicular to each other and be sure to use a WHOLE side length for your base. Wow, these new signs are going to be GREAT for Polygon's tourist industry! Just wait until word gets out that everyone is welcome here in Polygon...EVERYONE.