Perimeter and area on the coordinate plane
Perimeter and Area on the Coordinate Plane
Perimeter and Area on the Coordinate Plane is a topic in Two-Dimensional Measurement in the California Common Core State Standards. It is aligned to Standard G-GPE.7, which requires students to use coordinates to compute perimeters of polygons and areas of triangles and rectangles using the distance formula.
On the coordinate plane, side lengths come from the distance formula and areas from decomposing the figure.
Theory
On the coordinate plane, a polygon's measurements come from its vertices' coordinates:
- Side length from the distance formula \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
- Perimeter is the sum of the side lengths.
- Area by decomposing into rectangles and triangles, or by counting grid squares.
The distance formula:
How to measure a coordinate polygon
- Plot the vertices.
- Side lengths with the distance formula.
- Perimeter: add the side lengths.
- Area: split into rectangles/triangles and add.
Use the distance formula.
| \(d\) | \(=\) | \(\sqrt{(4-1)^2+(6-2)^2}\) |
| \(=\) | \(\sqrt{9+16}=\sqrt{25}=5\) |
The sides are \(4\) and \(3\); perimeter is twice their sum.
| \(P\) | \(=\) | \(2(4+3)=14\) |
Multiply the side lengths.
| \(A\) | \(=\) | \(4\times 3=12\) |
Base \(6\) along the \(x\)-axis, height \(4\).
| \(A\) | \(=\) | \(\dfrac12(6)(4)=12\) |
Common pitfalls
Frequently asked questions
How do you find a side length on the coordinate plane?
Use the distance formula between the two endpoints.
How do you find the perimeter of a coordinate polygon?
Add the lengths of all its sides, found with the distance formula.
How do you find the area of a coordinate polygon?
Decompose it into rectangles and triangles and add their areas, or count grid squares.
Which sides are easiest to measure?
Horizontal and vertical sides — their lengths are just the differences in the coordinates.