Area of a Rhombus: Diagonals, Height and Properties

A rhombus is a quadrilateral with four equal sides. Its opposite sides are parallel and its opposite angles are equal. To find its area, use either the two diagonals or a base and its perpendicular height. The side length alone is not enough.

How do the diagonals give the area?

Let D and d be the lengths of the two diagonals. Multiply them and divide by two. If a base b and its corresponding height h are known, their product gives the same area:

Both diagonals must use the same length unit. The answer uses the corresponding square unit. The letters D and d identify the diagonals; the calculation works even when they have equal lengths.

An example with diagonals of 10 cm and 6 cm

The product of the diagonals is 60 square centimetres. Taking half gives:

Using diagonals of 10 cm and 6 cm, step by step

Find the area of a rhombus with full diagonals of 10 cm and 6 cm. The diagonal formula uses their complete lengths, not their halves.

1. Substitute both diagonal lengths and retain the division by two.

   $$A=\frac{10\times 6}{2}$$

2. Multiply 10 by 6, then halve the product. The result is measured in square centimetres.

   $$A=\frac{60}{2}=30{\text{cm}}^2$$

3. Check geometrically: the half-diagonals measure 5 cm and 3 cm. Each of the four right triangles has area 7.5 square centimetres.

   $$A_1=\frac{5\times 3}{2}=7.50{\text{cm}}^2$$
   $$A=4\times 7.50=30{\text{cm}}^2$$

The four triangle areas sum to 30 square centimetres, matching the diagonal formula. Omitting the division by two would give twice the correct area.

Do not multiply the diagonals without dividing by two. The diagram shows why: a rhombus fills half the rectangle whose side lengths are its two diagonals, when its vertices lie at the midpoints of the rectangle's sides.

Deriving the formula from four triangles

The diagonals of a rhombus bisect each other at right angles. They split the figure into four right triangles. Each triangle has legs equal to half of D and half of d. Its area is half the product of those legs. Adding all four triangles gives half the product of the complete diagonals.

Alternatively, one diagonal divides the rhombus into two congruent triangles. Use that diagonal as their common base; the two heights together equal the other diagonal. This leads to the same expression.

Can you multiply the side by itself?

Only for a square. In a slanted rhombus, the perpendicular height is shorter than the side. The parallelogram formula multiplies base by perpendicular height, not by the sloping neighbouring side. Two rhombi may have equal side lengths but different areas because their angles differ.

If both diagonals are known, half-diagonals and a side form a right triangle. Pythagoras can recover the side and hence the perimeter:

For diagonals 10 cm and 6 cm, a side has squared length 34 in square centimetres. This is consistent with the 30-square-centimetre area; the area does not equal the side squared.

Is a square also a rhombus?

Yes. A square satisfies the four-equal-sides definition and has four right angles. Its diagonals are equal, unlike those of a non-square rhombus. In every rhombus, the diagonals also bisect the vertex angles, and an inscribed circle touches all four sides. Opposite angles are equal, and adjacent angles add to 180°. See the square's area and diagonal for this special case.