Area of Triangle: The Shoelace Formula!

Tired of complicated triangle area formulas? Did you know there's a super-fast way to find the area using just the coordinates of the vertices?

Subject: geometry • Classes: 6–12 • Difficulty: intermediate (9-10)

The Trick

The Shoelace Formula lets you find the area of a polygon (especially triangles) given its vertices $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$. For a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the area is: $\frac{1}{2} |(x_1y_2 + x_2y_3 + x_3y_1) - (y_1x_2 + y_2x_3 + y_3x_1)|$. Arrange the coordinates in a column, repeat the first coordinate at the end, multiply diagonally down to the right, sum them. Then multiply diagonally down to the left, sum them. Subtract the second sum from the first, take the absolute value, and multiply by $\frac{1}{2}$. It's called "shoelace" because of the criss-crossing multiplication!

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