Rhombus

In geometry, a rhombus (pl.: rhombi or rhombuses) is a special type of quadrilateral. A quadrilateral is a shape with four sides. In a rhombus, all four sides are the same length.

Other names for a rhombus include diamond, lozenge, and calisson.

A rhombus is a simple polygon, meaning it does not cross itself. It is a special kind of parallelogram and a kite. If the angles of a rhombus are all right angles, it becomes a square. If it is not a square, it has two smaller angles and two larger angles.

Etymology

The word rhombus comes from an ancient Greek word, ῥόμβος (rhómbos), which described something that spins, like a special spinning toy called a bullroarer. Famous thinkers like Euclid and Archimedes also used this word.

We call this shape a diamond because it looks like a shiny gem called an octahedral diamond. In card games, the diamonds symbol gets its name from this shape too.

The word lozenge might have come from a kind of sweet pastry or from the shape of old tombstones. And calisson is a French sweet that has the same rhombus shape.

Characterizations

A rhombus is a special four-sided shape where all sides are the same length. It is also called a diamond or a lozenge.

A four-sided shape is a rhombus if it meets any of these conditions:

• It is a special kind of shape called a parallelogram where one of its lines connecting opposite corners cuts an inside angle exactly in half.
• It is a parallelogram where at least two sides next to each other have the same length.
• It is a parallelogram where the lines connecting opposite corners meet at right angles.
• It is any four-sided shape where all four sides are the same length.
• It is a four-sided shape where the lines connecting opposite corners meet at right angles and cut each other exactly in half.
• It is a four-sided shape where each line connecting opposite corners cuts two opposite inside angles exactly in half.
• It is a four-sided shape where there is a special point so that four smaller shapes made from this point and the sides are all exactly the same size.
• It is a four-sided shape where certain smaller shapes inside it all share one common point.

Basic properties

Every rhombus has two diagonals that connect opposite corners, and two pairs of parallel sides. This makes a rhombus symmetric across each diagonal. Some key facts about rhombuses are:

• Opposite angles in a rhombus are equal.
• The two diagonals of a rhombus are perpendicular.
• The diagonals split the opposite angles in half.

A rhombus is a special type of parallelogram, meaning it has parallel opposite sides and angles that add up to 180 degrees.

Every rhombus is also a kite and has a circle that just touches all four sides.

Diagonals

The lengths of the diagonals in a rhombus can be found using the length of its sides and one of its angles. These formulas come from a basic rule in geometry called the law of cosines.

Inradius

The inradius is the radius of a circle that fits perfectly inside a rhombus. We can find this radius using the lengths of the rhombus's diagonals, called p and q, with the formula:

r = p ⋅ q / (2√(p² + q²))

We can also find the inradius if we know the length of a side, called a, and one of the angles, called α or β, using this formula:

r = (a · sin α) / 2 = (a · sin β) / 2

Area

For any parallelograms, the area of a rhombus is found by multiplying its base by its height. The base is one of its sides, called a.

You can also find the area using the side length and one of its angles. Another way is to use the lengths of the rhombus’s diagonals.

Dual properties

The dual polygon of a rhombus is a rectangle. A rhombus has all sides the same length, while a rectangle has all its angles the same.

A rhombus has opposite angles that are equal, while a rectangle has opposite sides that are equal. The figure made by joining the middle points of the sides of a rhombus is a rectangle, and the other way around.

Cartesian equation

A rhombus centered at the origin has special points called vertices. These vertices are at (±a, 0) and (0, ±b). This shape is a special type of a superellipse with an exponent of 1.

Other properties

One special pattern in two dimensions is called the rhombic lattice, also known as the centered rectangular lattice. Rhombuses can fit together to cover a flat surface in three different ways, including a special pattern called the rhombille tiling for rhombuses with 60-degree angles.

In three dimensions, shapes similar to rhombuses include the bipyramid and the bicone formed by spinning a rhombus around.

As the faces of a polyhedron

There are many solid shapes with rhombus faces. These include an endless group of shapes called rhombic zonohedrons, which are like stretched hypercubes.

A rhombohedron looks like a box (cuboid), but instead of rectangles, it has three pairs of rhombus-shaped faces. The rhombic dodecahedron is a solid with 12 identical rhombus faces. The rhombic triacontahedron has 30 special rhombuses called golden rhombi, where the lengths of the diagonals are in the golden ratio. The great rhombic triacontahedron is a complex solid with 30 intersecting rhombus faces. The rhombic hexecontahedron is formed by extending the rhombic triacontahedron and has 60 golden rhombus faces with a special kind of symmetry called icosahedral symmetry. The rhombic enneacontahedron is made of 90 rhombus faces, with three, five, or six meeting at each point. It has 60 wider rhombuses and 30 narrower ones. The rhombic icosahedron has 20 rhombus faces, with three, four, or five meeting at each point. It has 10 faces at the top and bottom and 10 around the middle.