Triangle: Area

The area of a shape is the size of the area inside the boundaries of that two-dimensional shape (once you start working with three dimensional shapes, area will refer to the two-dimensional area of one of the faces of that shape). 

To find the area of a triangle, you multiply the height times the length of the base, and then divide that answer by 2.

A few important things to think about: the height of the triangle must be perpendicular to the base.  Sometimes the height is one of the sides of the triangle (like the triangle to the right below).  Other times you have to draw in the height (called an altitude).  Sometimes, the altitude (the perpendicular line from the base of the triangle to the highest point of the triangle) is outside the triangle!  That's ok.  Just remember, the height is not necessarily a side.  Focus on finding the perpendicular line from the base to the highest point.

A trick to remember the formula for area of triangles.  Think about how, if you draw a diagonal line through a rectangle, you will form two triangles.  The height and base of each of those triangles will be the same as the length and width of the rectangle... but their areas will each only be half of that of the rectangle.  Voila!  That's why, to find the area of a triangle, we do $\dfrac{\text{base}\times\text{height}}{2}$. 

Units for area are always square units (e.g., square inches ($\text{in}^2$) or square feet ($\text{ft}^2$)).

Example: Find the area of the triangle below.

The base of the triangle is 6: $b=6$

The height of the triangle is 4: $h=4$

Note: One side of this triangle is 5, but that's not the height, so don't use it.  Tests often give you this measure to trick you!

$$\eqalign{\text{area}&=\dfrac{\text{base}\times\text{height}}{2}\\a&=\dfrac{6\times4}{2}\\a&=\dfrac{24}{2}\\a&=12 \text{ units}^2}$$

The only tricky part about finding the area of a triangle is figuring out what the height is -- and remembering to divide by 2 at the end!