Polygons on a coordinate grid
Draw vertices in order and use coordinate differences to measure sides.
Before you begin
- Coordinates in four quadrants
- Area and perimeter of rectangles
Draw the boundary in the given order
A list of vertices describes a polygon when adjacent points are joined and the last point is connected back to the first. Plot x first and y second using a consistent scale. Joining vertices in a different order can create crossing edges or a different shape. For a rectangle, check that opposite horizontal sides share y-values and opposite vertical sides share x-values.
Calculate lengths before area or perimeter
For a horizontal side, subtract its endpoint x-coordinates and take the absolute value; for a vertical side, use y-coordinates. This works even when the side crosses an axis. Once you have the lengths, calculate perimeter by adding boundary lengths and area using an appropriate formula or decomposition. The coordinates describe positions, not side lengths by themselves.
Worked example
A rectangle has vertices (−2, −1), (4, −1), (4, 3), (−2, 3). Find its perimeter and area.
Show the worked solution
- The horizontal length is |4 − (−2)| = 6 units.
- The vertical length is |3 − (−1)| = 4 units.
- Perimeter is 2(6 + 4) = 20 units; area is 6 × 4 = 24 square units.
Perimeter 20 units; area 24 square units.
Try it yourself
Which point completes the rectangle (−3, 1), (2, 1), (2, 5), ... in order?
Common mistakes
- Coordinates are positions; subtract endpoints to obtain side lengths.
- Join vertices in order and close the polygon.
What can you explain now?
A rectangle has corners (−4, −2), (1, −2), (1, 2), (−4, 2). Find its area and explain why using 4 × 2 would be wrong.
Compare with the explanation
20 square units.
Its width is 1 − (−4) = 5 and height is 2 − (−2) = 4. The coordinate magnitudes 4 and 2 measure distances from axes, not the full side lengths.
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Bring the ideas together
Area, surface area and volume
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Common Core mathematics standards ↗