Підхід до площі через координати з нуля

The area of a figure is the amount of space inside it.

For example, the area inside a square with side length 1 is $1^2 = 1$.

Now, there are a few handy formulas for areas of objects in the coordinate plane. One of them is Pick's formula.

Suppose we have a polygon with all the vertices on lattice points (points with integer coordinates).

If $B$ is the amount of points on the boundary, I is the amount of lattice points inside the figure, and A is the area, we have $$A=\frac{B}{2} + I -1$$

Example. Draw a square with coordinates $(0,0)$, $(0,2)$, $(2,0)$, and $(2,2)$.

Pick's formula tells us the area is $\frac{8}{2}+1-1=4$. This makes lots of sense.

We can draw the quadrilateral- Pick's formula tells us the answer is $\frac{4}{2}+23-1=24$.

Another well known theorem for finding the area of a polygon is the Shoelace Theorem.

Again, suppose we have a polygon P with all the vertices on lattice points (points with integer coordinates). Also suppose it has vertices $(a_1, b_1),(a_2, b_2),\cdots,(a_n, b_n)$, listed in clockwise order. Then the area of P is

$$\dfrac{1}{2} \cdot \left|(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)\right|$$

Areas

Definition

Areas of simple polygons

Pick's Theorem - for more complicated polygons

Applications

Exercises

checkin'