ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
Area and Perimeter on the Coordinate Plane
When you know a polygon's corner coordinates, you can get its exact area and perimeter with formulas instead of a ruler: use horizontal and vertical distances for area, and the distance formula for each side's length to total the perimeter.
Coordinates turn shape into arithmetic. Every side's length is a distance you can compute, and every height is just a vertical or horizontal gap between numbers. That means a figure's area and perimeter fall out of subtraction and the distance formula, no measuring required. The key move is choosing a base and height that line up with the grid. If two vertices share the same y-value, the segment between them is horizontal, and its length is just the difference in their x-values. The height is then how far the third vertex sits above or below that line, measured straight up or down.
When a triangle or quadrilateral is tilted so no side is flat, box it inside the smallest upright rectangle that contains it. The rectangle's area is easy: length times width, read straight off the grid. Each corner of the original figure cuts off a right triangle (or several) from that rectangle. Find those leftover pieces' areas and subtract them from the rectangle's total, and what remains is the area of the tilted figure. This boxing-and-subtracting trick works for any polygon, however irregular.
Perimeter asks a different question: not what's enclosed, but how far around. Here the distance formula √((x2-x1)2 + (y2-y1)2) does the work, applied once per side, then added up. This topic sits right after slope and midpoint because the same coordinate pairs feed all of it: slope tells you if sides are parallel or perpendicular, midpoint locates centers, and distance and area close out what a set of points can tell you about a shape.
Key terms
- distance formula
- √((x2-x1)2 + (y2-y1)2), the length between two coordinate points.
- base and height
- a side length and the perpendicular gap to the opposite vertex, used in area = (1/2)(base)(height).
- bounding rectangle
- the smallest upright rectangle enclosing a tilted figure, used to find area by subtracting corner triangles.
- perimeter
- the total distance around a figure, found by summing the lengths of all its sides.
Watch it explained
- How it worksHow does boxing a tilted triangle inside a rectangle let you subtract your way down to its area?
- Worked exampleFind the area of the triangle with vertices (1, 1), (7, 1) and (4, 6).
- Worked exampleFind the perimeter of the quadrilateral with corners (0, 0), (4, 3), (9, 3) and (5, 0).
- Common mistakeWhy doesn't counting whole grid squares inside a tilted figure give its exact area?
Check yourself
End of topic.