Shoelace Formula Magic!

Struggling to find the area of a triangle on a coordinate plane? Forget complex formulas involving height and base! Unleash the Shoelace Formula magic!

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

The Trick

The Shoelace Formula (or Surveyor's Formula) is your secret weapon! 1. List coordinates $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ of the triangle's vertices in counter-clockwise or clockwise order. 2. Repeat the first coordinate at the end: $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_1, y_1)$. 3. Calculate the sum of 'downward' diagonal products: $P = x_1y_2 + x_2y_3 + x_3y_1$. 4. Calculate the sum of 'upward' diagonal products: $N = y_1x_2 + y_2x_3 + y_3x_1$. 5. The Area $A = \frac{1}{2} |P - N|$. *WHY it works:* It's derived from the concept of signed area using determinants or by decomposing the polygon into trapezoids whose areas cancel out correctly, leaving the polygon's 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.