Rhombus and Kite

A leaning square

A rhombus is a quadrilateral whose four sides are all equal. A square has four equal sides as well, but the corners of a rhombus need not be right angles: it is a square that has been pushed over, so that its corners lean. A square is the rhombus whose angles are all 90°.

Every side of this rhombus runs 2 dots across and 3 dots up or down, so all four sides are the same length. None of its corners is a right angle.

Fold it along a diagonal

A diagonal joins two opposite corners. Fold the rhombus along the diagonal from the top corner to the bottom corner. The left corner is one side’s length from the top corner and one side’s length from the bottom corner, and folding does not change those distances. On the right of the diagonal there is only one point at those two distances, just as two arcs struck from the ends of a base cross at only one point on each side of it. The right corner is at exactly those distances, so the fold lands the left corner on the right corner, and the two halves of the rhombus match.

The other diagonal joins those two corners, so it is folded onto itself: its left part lands on its right part, and the diagonal is cut exactly in half. The two angles where the diagonals cross also land on each other, so they are equal, and they lie on a straight line, so each is 180° ÷ 2 = 90°.

All four sides are equal, so the other diagonal is a fold line too, and the same argument cuts the first diagonal in half. The diagonals of a rhombus cross at right angles, and each cuts the other in half.

The diagonals cross 3 dots below the top corner and 3 dots above the bottom one, and 2 dots from each side corner. One is upright and the other runs straight across, so they cross at a right angle.

The fold needs equal neighbors

The fold worked because each side corner is the same distance from the top as the other side corner, and the same distance from the bottom. Take a rectangle whose length and width are different, and fold it along a diagonal. One of the other two corners is a length away from the top end of the diagonal and a width away from the bottom end; the other corner is a width away from the top end and a length away from the bottom end. The distances are swapped, so the fold misses.

The rectangle folded corner to corner: the folded half sticks out past the edges, because the rectangle’s length and width are different. Choose the square, a rhombus with right angles, and the same fold lands exactly.

What else the folds give

Folding along a diagonal also lays one half of each corner angle it passes through on the other half. So each diagonal of a rhombus bisects the two angles at its ends: it cuts each of them into two equal angles.

Each fold lays one corner on the corner opposite it, so opposite angles are equal. Two pairs of equal angles that add to 360° means that any two neighboring angles add to 360° ÷ 2 = 180°. Two angles at the ends of a side that add to 180° are co-interior angles, so the sides at the other ends are parallel. A rhombus is a parallelogram with four equal sides, and every property of a parallelogram belongs to it.

The kite

A kite has two pairs of equal sides, and each pair sits side by side, meeting at a corner. In this kite the two short sides meet at the top and the two long sides meet at the bottom.

Fold the kite along the diagonal from the top corner to the bottom corner. Each side corner is a short side away from the top and a long side away from the bottom, so the fold lands one side corner on the other, exactly as in the rhombus. The kite folds flat along this diagonal, which makes it a line of symmetry: the angles at the two side corners are equal, and the angles at the top and the bottom are each cut in half.

The two short sides run 2 dots across and 2 down from the top corner. The two long sides run 2 dots across and 4 up from the bottom corner. Equal sides meet in pairs.

Only the fold line halves the other

The fold lays the left corner on the right corner, so, just as in the rhombus, the diagonal across is cut in half, and the two diagonals cross at right angles.

The fold line itself is not cut in half. Nothing lays its top part on its bottom part, and in this kite they are different lengths: the crossing is 2 dots below the top corner and 4 dots above the bottom corner. The diagonal across is not a line of symmetry either, and it does not cut the side corners in half.

A rhombus is the kite whose four sides are all equal. That is why both of its diagonals are fold lines, and both are cut in half.

The diagonal across is cut in half, 2 dots on each side of the upright one. The upright diagonal is not: it is 2 dots above the crossing and 4 dots below it.

Area from the diagonals

Because the diagonals cross at right angles, they cut a rhombus or a kite into four right-angled triangles. Draw a rectangle around the shape, with a side through each corner. The rectangle is as long as one diagonal and as wide as the other, and the diagonals cut it into four smaller rectangles. Each of the four triangles is half of one of those small rectangles, so the shape is half of the big rectangle: area $=d_1\times d_2/2$= d₁ × d₂ / 2, where $d_1$d₁ and $d_2$d₂ are the lengths of the diagonals.

This kite’s diagonals are 4 and 6, so its area is 4 × 6 ÷ 2 = 12 square units. Check by triangles: the diagonal across splits it into a top triangle with base 4 and height 2, area 4, and a bottom triangle with base 4 and height 4, area 8, and 4 + 8 = 12.

The rectangle around the kite is 4 by 6, which is 24 squares. The diagonals cut it into four smaller rectangles, and the kite takes half of each, so its area is 24 ÷ 2 = 12.

Two slips

The right angle is where the diagonals cross, not at a corner of the shape. A kite’s corners can be any size; only the angle between its diagonals is sure to be 90°.

Only a rhombus has diagonals that bisect all four of its angles. In a kite only the fold line bisects its angles, and in a parallelogram that is not a rhombus neither diagonal does.