Cut a Triangle into a Rectangle — ½ base × height
Why is a triangle ½ base × height? Cut along the midline, rotate the top pieces, and a clean rectangle appears — the formula made visible.
Problem
Turn any triangle into a rectangle using only cut-and-rotate, showing its area is ½ × base × height.
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.)
Any triangle (base b, height h). Cut along the midline joining two side-midpoints.
Why this approach works
Cut horizontally at half the height — the midline joining two side-midpoints. Split the top triangle down the middle and rotate each half 180° about a midpoint; they fit the slanted sides of the trapezoid exactly, raising vertical walls. The rectangle has the same base b and half the height, so the triangle = b × (h/2) = ½ b h, for any triangle.
Proof
The two halves of the upper triangle rotate to fill the trapezoid into a base-b, height-h/2 rectangle. Area is preserved, so the triangle equals ½ b h.
Try it yourself
Imagine an obtuse triangle: the two top halves change shape, but rotated 180° they always tile the same base × (height/2) rectangle.
Related mathematics
Beyond MathVoyage
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