The Bird's Head Model: Find This Area in 10 Seconds

Step by step

1. A big triangle: area = 24
   Here's a big triangle. Its area is twenty-four.
2. Cut off the tip → a bird's head
   Cut off the tip with a line, and you get a small triangle that looks like a bird's head.
3. Left 2 + 1, right 3 + 1. Head = ?
   The cut splits the left side into two and one, and the right side into three and one. How big is the bird's head?
4. Same corner → areas compare like bases
   We need one tool. When two triangles share the same top corner, and their bases sit on one line, their areas compare like their bases. Bases of two and three give areas of two and three.
5. The blue triangle is our bridge
   Now draw this line. It makes a blue triangle in the middle. That's our bridge.
6. Bases 3 and 4 → head = 3/4 of blue
   The bird's head and the blue triangle share this corner. Their bases are three and four, so the head is three quarters of the blue triangle.
7. Bases 2 and 3 → blue = 2/3 of whole
   The blue triangle and the whole triangle share this corner. Their bases are two and three, so the blue triangle is two thirds of the whole.
8. 3/4 × 2/3 = 6/12 = 1/2 → 1/2 × 24 = 12
   Put it together. Three quarters of two thirds is six twelfths, which is one half. Half of twenty-four is twelve!
9. Shared angle → multiply the side ratios
   That's the bird's head model. When two triangles share an angle, multiply the side ratios. Two thirds times three quarters is one half.