Which Triangle Is Biggest? Your Eyes Will Lie to You

Step by step

1. Which triangle is biggest? Pick one!
   Which of these three triangles has the biggest area? Pick one before we measure.
2. All the same! Same bottom, tops on one line
   Surprise: they're all the same. They share the same bottom, and their tops sit on the same straight line.
3. Way 1: base × height ÷ 2 → same base, same height
   Way one is the school formula. Area is base times height, divided by two. All three have the same base and the same height, so they have the same area.
4. Way 2: a triangle as a stack of thin cards
   Way two shows why the slant doesn't matter. Build a triangle out of thin cards, like a stack of playing cards.
5. Slide the cards → same cards, same area
   Now push the stack sideways. Each card just slides, and no card changes size. So the area stays the same, while the top slides along the line.
6. Same base + same height → same area
   So triangles with the same base and the same height always have the same area. In China, this is the start of the same-height model.
7. Shared top corner → same height · bases 2 and 3
   Now the main idea. These two triangles share the top corner, so they have the same height. Their bases are two and three.
8. 5 equal steps → 5 equal thin triangles
   Cut the bottom into five equal steps. Each thin triangle has the same base and the same height, so they're all equal.
9. Same height → areas compare like the bases
   So the left triangle is two pieces, and the right one is three. With the same height, areas compare just like the bases!
10. Bottom 4 ft and 6 ft · left bed 10 sq ft · right bed?
    Here's a real one. A path from the top corner splits a garden's bottom into four feet and six feet. The left bed is ten square feet. How big is the right bed?
11. 4 : 6 = 2 : 3 · 1 piece = 5 · right = 15 sq ft
    Four to six is two to three. Ten is two pieces, so one piece is five. The right bed is three pieces: fifteen square feet!
12. On a test: same height · 4 × h ÷ 2 = 10 → h = 5 → 6 × 5 ÷ 2 = 15
    On a test, write it like this. Both beds share the top corner, so they have the same height. Use area equals base times height, divided by two. The left bed gives a height of five. So the right bed is six times five, divided by two: fifteen square feet.