Area puzzles

Many area puzzles come down to one idea: triangles on the same base, between the same parallels, have the same area. Slide the top corner along the parallel line: the height stays the same, so the area does too.

Parallelogram ABCD with P and Q on side AB: triangles PCD and QCD share base CD and lie between the same parallels, so their areas are in the ratio 1 : 1.

A median, used twice. D is the midpoint of BC and P is on AD:

• ABD and ACD are equal (median AD);
• PBD and PCD are equal (median PD);
• subtract: ABP = ACP.

A point inside a square. Join P to the corners. The heights of triangles PAB and PCD add up to the side, so together they make half the square; so do the other two.

The midpoint 4-gon. Joining the midpoints of a 4-gon's sides cuts off four corner triangles, each a quarter of the triangle it sits in. Together they remove half the 4-gon, so the middle parallelogram is half its area.

Rhombus area = ½ × d₁ × d₂. With diagonals d and 2d and area 128: d² = 128, so the shorter diagonal is 8√2 ≈ 11.3 cm.