The Shoelace Area Trick!

Ever struggled to find the area of a triangle when you only have its vertices' coordinates? Forget complex determinants! Let's 'tie' up the points to get the answer super fast!

Subject: geometry • Classes: 6–12 • Difficulty: intermediate

The Trick

The Shoelace Formula (or Surveyor's Formula) is a gem! List the coordinates $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ vertically, repeating the first point at the end. Then, sum the diagonal products downwards (left to right) and subtract the sum of diagonal products upwards (right to left). Take the absolute value and divide by 2. $A = \frac{1}{2} | (x_1y_2 + x_2y_3 + x_3y_1) - (y_1x_2 + y_2x_3 + y_3x_1) |$ Why it works: It's a clever way to compute the signed area, which can be seen as the sum/difference of areas of trapezoids formed by projecting the vertices onto an axis, or directly related to the determinant formula for area.

Study More with GyanAI

GyanAI is a free AI tutor for CBSE students. Ask any question for an instant step-by-step answer. Try GyanAI free.