Area of a Triangle on a Square Grid
What is the area of the green and the blue triangle below?
If we look carefully, we notice that their area is exactly half of the rectangle into which they could fit:
The red rectangle has an area of 6 squares. The green and the blue triangle each have exactly half of that, so their area is 3 squares.
This rule works for every triangle. In the picture below, the green and the blue triangle also have an area equal to half of their rectangle — again 3 squares:
And even when the vertex of the triangle “sticks out” of the rectangle, its area is still half of it. That is why all the triangles in the picture below have the same area — half the area of the red rectangle:
So when we find the area of a triangle on a square grid, it does not have to be difficult at all. We just need to find the right rectangle or square that “belongs” to it. It is not a coincidence that the formula for the area of a triangle is: base times height to that base, all divided by two. By height we mean the perpendicular distance from the opposite vertex to the given side. The marked height of the purple triangle is 1 (so its area is 1 square), for the green triangle it is 2 (so its area is 2 squares), and for the blue triangle it is 3 (so its area is 1 and a half squares).
However, this method can only be used when one side of the triangle lies directly on the grid lines. But what about triangles that do not have such a side, like triangle ABC below?
