When a figure's vertices are given as coordinates, the distance formula from Distance and Midpoint Formulas finds every side, so you can find its perimeter. For area, a rectangle drawn around the figure turns a slanted shape into pieces you already know.
Por qué funciona
Sides along grid lines are easy: count. Slanted sides come from the Pythagorean theorem, so they are often square roots. Only like radicals combine: $\sqrt{8} + \sqrt{2} = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$, but $\sqrt{5} + \sqrt{13}$ stays as it is. For area, the box method works because the box is a rectangle and the leftover corner pieces are right triangles (and rectangles) whose legs lie along the grid.
Idea clave
Perimeter: add the side lengths from the distance formula, simplifying radicals and combining like ones. Area: enclose the figure in a grid-aligned rectangle and subtract the right triangles and rectangles in the corners.
Ejemplo resuelto: A perimeter with radicals
Find the perimeter of the triangle with $A(1, 1)$, $B(7, 1)$ and $C(3, 5)$.
- $AB = 6$ (horizontal).
- $BC = \sqrt{(3 - 7)^2 + (5 - 1)^2}$ $= \sqrt{32} = 4\sqrt{2}$.
- $CA = \sqrt{(1 - 3)^2 + (1 - 5)^2}$ $= \sqrt{20} = 2\sqrt{5}$.
Respuesta: $6 + 4\sqrt{2} + 2\sqrt{5} \approx 16.13$
Ejemplo resuelto: The box method
Find the area of the triangle with $A(1, 1)$, $B(7, 2)$ and $C(3, 6)$.
- The box from $x = 1$ to $7$ and $y = 1$ to $6$ is $6 \times 5 = 30$.
- Corner triangles: $\frac{1}{2} \cdot 6 \cdot 1 = 3$ (below $\overline{AB}$), $\frac{1}{2} \cdot 4 \cdot 4 = 8$ (right of $\overline{BC}$), $\frac{1}{2} \cdot 2 \cdot 5 = 5$ (left of $\overline{CA}$).
- Area $= 30 - 3 - 8 - 5 = 14$.
Respuesta: $14$ square units
Ejemplo resuelto: A right triangle on the grid
Find the area and perimeter of the triangle with $P(-2, -1)$, $Q(4, -1)$ and $R(-2, 7)$.
- $\overline{PQ}$ is horizontal (length 6) and $\overline{PR}$ is vertical (length 8): a right angle at $P$.
- Area $= \frac{1}{2} \cdot 6 \cdot 8 = 24$. $QR = \sqrt{36 + 64} = 10$, so the perimeter is $24$.
Respuesta: Area $24$, perimeter $24$
Error común
Adding unlike radicals, like $\sqrt{32} + \sqrt{20} = \sqrt{52}$. Square roots do not add that way: $\sqrt{32} + \sqrt{20} \approx 5.66 + 4.47 = 10.13$, but $\sqrt{52} \approx 7.21$. Simplify each radical, then combine only the ones with the same number under the root.