Connect the Midpoints — Exactly Half
Join the midpoints of a rectangle to form a rhombus. No matter the proportions, the rhombus is always exactly half. Why?
Problem
Joining the four side-midpoints of a rectangle forms a rhombus. What fraction of the rectangle is the rhombus?
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
Join the four midpoints into a rhombus. What is its area?
Why this approach works
Don't compute the ratio — watch the leftover pieces. The four corner right-triangles each rotate 180° about a side-midpoint, and all four apexes land at the rectangle's center, exactly tiling the rhombus. So the four outer pieces equal the inner rhombus; together they make the whole rectangle. The rhombus is exactly half — independent of shape.
Proof
The four corner triangles rotate to tile the rhombus exactly, so the rhombus equals half the rectangle — independent of proportions.
Try it yourself
Imagine a very flat rectangle, or a square. The corner triangles change shape, but four of them always tile the rhombus — the ratio never moves off ½.
Related mathematics
Beyond MathVoyage
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