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Area of a Parallelogram
Rung 1 of 4 · Discover

Where the Parallelogram Formula Comes From

Here's the lovely secret: a parallelogram is just a rectangle that's been pushed over. Cut a triangle off one end, slide it to the other, and you've got the rectangle back — same area the whole time.

NESA MA4-ARE-C-01Builds on the rectangle

PlayDrag the slant to lean it over, then hit “straighten into a rectangle” and watch the area hold steady.
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Audio WalkthroughComing Soon
Video ExplainerComing Soon

A parallelogram looks trickier than a rectangle because it leans. But that lean is a lie — there's a clean rectangle hiding inside it, and finding it is the whole trick.

The Cut-and-slide

Take the parallelogram and slice straight down from the top corner, making a right-angle cut. That chops a triangle off one end. Now slide that triangle across to the other end, where it fits perfectly into the gap. What's left? A plain rectangle. You moved a piece around but you didn't add or remove any space, so the area is exactly the same. Press "straighten into a rectangle" in the toy and you'll watch it happen.

Say it plainly: a parallelogram is a rectangle in disguise. Its area = base × perpendicular height — the same base, and the straight-up height, not the slanted side.

So the Formula Is the Rectangle's Formula

The rectangle you straightened it into is base wide and height tall, so its area is base × height. Since the parallelogram has the same area, it must be base × height too. No new rule to learn — it's the rectangle formula wearing a hat.

Why "perpendicular" Height Matters

The height has to be measured straight up from the base, at a right angle — because that's the height of the hidden rectangle. The slanted side is longer, and it's the wrong number. We'll hammer that home on rung 3, but notice it now: dragging the slant makes the side longer while the upright height never budges.

Us, Thinking Out Loud

Why does moving the triangle across not change how much space there is?

Could you cut and slide it the other way and still get a rectangle?