Where Does "÷ 2" in the Triangle Area Formula Come From?

Step by step

1. No formula: a triangle field, 10 wide, 6 tall. Area?
   No formula allowed. This triangle-shaped field is ten units wide and six tall. How big is it?
2. Liu Hui (China, 263 AD): cut, move, same area
   About seventeen hundred and fifty years ago, Liu Hui wrote notes on an old Chinese math book, the Nine Chapters. His big idea: cut a shape, move the pieces, and the area stays the same.
3. Slanted box: cut a corner, slide it → base × height
   Start with the easiest field: a slanted box. Cut this corner off, and slide it to the other side. Now it's a rectangle, so its area is base times height.
4. Triangle: the middle of each slanted side, a line down
   Now our triangle. Find the middle of each slanted side, and draw a line straight down from each one.
5. Half a turn around each middle point → fills the gap
   Swing each bottom corner piece half a turn around its middle point. It fits right into the gap at the top.
6. A rectangle: half as wide, just as tall → 5 × 6 = 30
   Now the triangle is a rectangle. It's just as tall, but half as wide. Half of ten is five, and five times six is thirty!
7. Half the width × the height (the Nine Chapters)
   That's where divided by two comes from. The Nine Chapters says it the same way: half the width, times the height.
8. Two parallel sides: swing the corners the same way
   The same trick works for a field with two parallel sides. Swing the corner pieces the same way.
9. Width = halfway between the parallel sides → average × height
   Now it's a rectangle, and its width is halfway between the two parallel sides. So the area is the average width times the height.
10. Cut, move, same area
    Cut, move, same area. One idea gives the area of all three fields.
11. On a test: rectangle 5 × 6 = 30 = 10 × 6 ÷ 2
    On a test, write it like this. Cut and move: the triangle becomes a rectangle five wide and six tall. Its area is five times six. That's the formula: ten times six, divided by two. The field is thirty square units.