Pick's Theorem -- Page 2

Consider the following graphic.

The grid of dots above is called "dotpaper". We assume each dot is 1 unit from the next dot horizontally (accross) or vertically (up or down). Of course, dots diagonally next to each other are farther appart. The square A above is the "unit square", its area is 1. The four triangles B form half a unit square, so they have area 1/2.

These are some rectangles. You can find their area by counting the unit squares they contain. So A has area 8, B has area 9 and C has area 3. Alternatively one can measure the base and height and multiply, for example figure A has base 4 and height 2, so area = 4*2 = 8. For B the base and height are both 3 so the area is 3*3 = 9 while the last rectangle has base 1 and height 3 so its area is 1*3 = 3. In general the formula for the area of a rectangle is A = bh where b is the base (horizontal) and h is the height (vertical).

We can use the formula A = 1/2 bh for a triangle. In triangle A then the base is horizontal of length 2 and the height is vertical of length 3 so the area is 1/2(2)(3) = 3. For B the base is 4 and the height h is the vertical distance from the opposite vertex to the base, mainly 2 so A = 1/2 (4)(2) = 4. Triangle C is similar but here we have the base in the vertical direction, length 3, and the height is the horizontal distance from the opposite vertex to the base which is 1. So A = 1/2(3)(1) = 3/2. Finally in triangle D the horizontal base is 4 and the vertical height is 2, here it is the distance from the vertex opposite the base to the horizontal line of the base. Again the area is 4.

Remember the important thing in dot figures is that we can only use horizontal or vertical distances, never slant distances.

Now you try one.

(The buttons here are just for display, but indicate a possible answer which you can mark. But there will be no action taken.)

Find the area of the triangle above:
4
6
9
18