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Visual geometry
Let your eyes be wrong firstMotion becomes proof

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.

Do not write an equation yet. Shake the figure first.

Symmetry & slidingDifficulty · Easy

Play the reversal slowly

Now move the same pieces step by step and prove that the pattern you touched was not an accident.

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base bheight h

Any triangle (base b, height h). Cut along the midline joining two side-midpoints.

Return to the exact question

Turn any triangle into a rectangle using only cut-and-rotate, showing its area is ½ × base × height.

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

Area = ½ × base × height

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.

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