Area of a Triangle

This image is a visual demonstration of the fact that the area of a triangle is one half of the base times its height.  Within the overall rectangle above, there are three main triangles of interest.  There's a blue one, made up of the two darker blue regions on the bottom plus the lighter blue region on the right side.  This right triangle has the same height as the overall rectangle and its base is the length of the overall rectangle.  Since the hypotenuse of the right triangle splits the rectangle into two equal areas, the area of the blue triangle is half the area of the rectangle, or A = 1/2 × base × height.

Alternatively, there's an acute triangle, made up of the two darker blue pieces plus the two darker yellow pieces on top of them.  This triangle also has a height that's the same as the overall triangle.  The altitude (white line from the bottom of the triangle to the top point) splits the overall rectangle into two smaller rectangles.  Each smaller rectangle has two smaller right triangles, each with one yellow piece and one blue piece, and both right triangles have areas that are half of that of their respective rectangles.  So, the area of the left small yellow & blue triangle (which is half of the area of its rectangle) + the area of the right small yellow & blue triangle (which is half of the area of its rectangle) = the area of the large yellow & blue triangle, which is half of the area of the overall rectangle.

Finally, there's the case of an obtuse triangle, played here by two darker yellow pieces plus the lighter yellow piece.  The big yellow triangle plus the green one together make up the other half of the overall rectangle (across the diagonal from the blue one).  By itself, the green triangle makes up half of the smaller left rectangle.  Therefore, the area of the yellow triangle has to be half of the smaller right rectangle, since the smaller left + smaller right rectangles make up the overall rectangle.  The area of the smaller right rectangle is its height (same as the height of the yellow triangle) × its width, which is the same as the base of the yellow triangle.  Once again, this shows that A = 1/2 × base × height.