Put a polygon on the coordinate plane and its side lengths come from the coordinates. Use distances on the grid for sides, and area ideas for the inside.
Key idea
Plot the vertices and join them in order. A side along a grid line has length equal to the distance between the coordinates that change. Use those lengths for perimeter and area.
Why it works
Moving from A to B changes only x, from $-3$ to 4, so AB is horizontal and its length is the distance between $-3$ and 4. That is the same as walking along the x-axis: 3 units to 0, then 4 more. The coordinates turn geometry into number lines you already know.
Worked example: Perimeter and area
Rectangle ABCD has vertices $A(-3, -2)$, $B(4, -2)$, $C(4, 3)$ and $D(-3, 3)$. Find its perimeter and area.
- AB: from $x = -3$ to $x = 4$, so $3 + 4 = 7$.
- BC: from $y = -2$ to $y = 3$, so $2 + 3 = 5$.
- Perimeter: $2 \times 7 + 2 \times 5 = 24$. Area: $7 \times 5 = 35$.
Answer: Perimeter 24 units, area 35 square units.
Worked example: The missing vertex
Three vertices of a rectangle are $(-2, 1)$, $(5, 1)$ and $(5, -4)$. Find the fourth.
- Each x-coordinate and each y-coordinate of a rectangle like this is used twice.
- x: $-2$ is used once. y: $-4$ is used once.
Answer: $(-2, -4)$
Worked example: A right triangle
Find the area of the triangle with vertices $(-2, 1)$, $(3, 1)$ and $(-2, 5)$.
- The horizontal side has length $2 + 3 = 5$ and the vertical side has length $5 - 1 = 4$.
- They meet at a right angle at $(-2, 1)$: $A = \frac{1}{2} \times 5 \times 4 = 10$.
Answer: 10 square units
Common mistake
Subtracting coordinates on opposite sides of 0 as if they were on the same side: the side from $x = -3$ to $x = 4$ is 7, not 1. Quick check: count the grid squares along one side.