# Area of a rhombus

The area of a rhombus is the space contained within its perimeter. A rhombus is a parallelogram in which all sides are congruent. The grey space is the area of the rhombus in the diagram below.

## Area formula

There are several formulas that can be used to find the area of a rhombus depending on the known parameters.

### Using diagonals

The area, A, of a rhombus is half the product of its two diagonals.

where d_{1} and d_{2} are the lengths of the two diagonals.

Referencing rhombus ABCD above, Let AC = d_{1} and BD = d_{2}. The diagonals of a rhombus are perpendicular to each other, so AC⟂ DB. Rhombus ABCD can be decomposed into triangles ABC and ADC.
The area of △ABC = where BE is the altitude of △ABC. The area of △ADC = where DE is the altitude of △ADC. The area of rhombus ABCD equals the sum of the areas of △ABC and △ADC.

△ABC + △ADC = | |

= | |

= | |

= |

Example:

If the area of a rhombus is 230, and one of its diagonals is 10, what is the length of the other diagonal?

Let d_{1} = 10. Since A = 230 we can find d_{2} as follows:

230 = 5 × d_{2}

d_{2} = 46

### Using side and height

Since a rhombus is also a parallelogram, we can use the formula for the area of a parallelogram:

A = b×h

where b is the base or the side length of the rhombus, and h is the corresponding height.

### Using side and angle

If the side length and one of the angles of the rhombus are given, the area is:

A = a^{2} × sin(θ)

where a is side length and θ is one of the angles.

## Finding area using a grid

Another way to find the area of a rhombus is to determine how many unit squares it takes to cover its surface. Below is a unit square whose dimensions are 1 cm.

A grid of unit squares can be used when determining the area of a rhombus.

The grid above contains unit squares that have an area of 1 cm^{2} each. The rhombus on the left contains 8 full squares and 12 partial squares, so it has an area of approximately:

The rhombus to the right contains 25 full squares, so it has an area of approximately 25 cm^{2}.

This method can be used to find the area of any shape; it is not limited to rhombuses. However, it is only an approximate value of the area. The smaller the unit square used, the higher the accuracy of the approximation. Using a grid made up of 1 mm squares is 10 times more accurate than using a grid made up of 1 cm squares.