# High School: Geometry

### Expressing Geometric Properties with Equations HSG-GPE.B.7

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

If there's one thing that mathematicians love, it's efficiency. And the distance formula is nothing if not efficient. It is so efficient that while we're going the distance, we can even go for speed.

Students are expected to be able to use the distance formula to find the distance between two coordinate points and then apply that information and know-how to calculate the perimeter and area of various polygons.

They should note that the distance formula is derived from the Pythagorean theorem. We suggest squaring both sides of the distance formula so it's a little easier to see that both equations set the sum of two squared terms equal to a third squared term.

Basically, the distance formula assumes that the distance we're measuring is the hypotenuse of a right triangle. The base of the triangle is denoted as a, or (x2x1). The height is b, or (y2 – y1). And the hypotenuse, c, is the distance, D. Now that that's all cleared up, let's go watch some E!

As one might expect, the distance formula is pretty handy for computing the distance between two coordinate points. Of course, it's best saved for when those points are not located along the same vertical or horizontal, because then simple subtraction would be the most efficient way to find the distance between them. No, the distance formula is more aptly used when the points are spaced diagonally.

Students should know that if we compute the distances between the points around a polygon, then the distances can be added to find the polygon's perimeter. Students should know that this works for all polygons.

Area is a little different, since it depends on the actual shape. Students should know that pretty much all polygons on the coordinate plane can be split into rectangles and triangles. As such, students should know how to calculate the areas of rectangles and triangles. Adding the areas of the individual pieces should give them the area of the entire shape.

We recommend giving students a variety of shapes so they can apply their knowledge and problem-solving skills rather than automatically plugging points into a formula. If students are ever confused, they can always plot points on the coordinate plane, too. After all, this is still geometry, and a picture can speak volumes (er…areas).

Here isa recap video on coordinate and distance formula.