Where Do Area Formulas Come From?

Step by step

1. Where do area formulas come from?
   What is area, really? And where do formulas like length times width come from? Let's build them all from one simple idea.
2. Area = how many unit squares fit inside
   Area is the amount of flat space a shape covers. We measure it with unit squares: squares that are one unit on each side. Each one is one square unit, like one square centimeter.
3. 4 by 3 rectangle → 12 unit squares
   Take a rectangle, four units long and three units wide. Fill it with unit squares, and count: twelve. So its area is twelve square units.
4. 3 rows of 4 → Area = length × width
   But we don't need to count one by one. There are three rows, with four squares in each row. Four times three is twelve. That's why the area of a rectangle is length times width.
5. Square: side × side = side²
   A square is just a rectangle with equal sides, so its area is side times side: side squared.
6. Parallelogram: base × height
   Now, a parallelogram. Cut off this triangle, and slide it to the other side. It's a rectangle! Same base, same height. So the area of a parallelogram is base times height.
7. Height: straight up, not slanted
   Careful: the height goes straight up, at a right angle to the base. It's not the slanted side.
8. Triangle: ½ × base × height
   What about a triangle? Two copies of the same triangle fit together into a parallelogram. So one triangle is half of it: one half, times base, times height.
9. Trapezoid: ½ × (a + b) × height
   A trapezoid works the same way. Two copies make a parallelogram whose base is a plus b. So the area is one half, times a plus b, times the height.
10. Circle → πr × r = πr²
    Even a circle! Cut it into thin slices, and line them up, one pointing up, one pointing down. It looks almost like a rectangle, and the thinner the slices, the closer it gets. Its base is half the circumference, pi r, and its height is the radius, r. So the area is pi r times r: pi r squared.
11. Cut & move pieces: same area
    So every formula comes from one idea: area counts unit squares, and cutting and moving pieces never changes the area.
12. Quick check: base 6, height 4. Area?
    Quick check: a triangle with base six and height four. What's its area?
13. ½ × 6 × 4 = 12 square units
    One half, times six, times four: twelve square units! Which shape should we explain next? Tell us in the comments.