How to Calculate the Area of a Rhombus

A rhombus is a specific type of quadrilateral with all sides equal and opposite sides parallel, exhibiting unique geometrical properties that enable the calculation of its area through several distinct methods involving base and height, diagonals, or the measurement of included angles.

Formal Definition and Geometrical Structure of a Rhombus

A rhombus is defined as a convex quadrilateral where all four sides are congruent. Let $ABCD$ be a rhombus with vertices in order. By definition, $AB = BC = CD = DA$ and $AB \parallel CD$, $BC \parallel DA$. In a rhombus, the two diagonals $AC$ and $BD$ intersect at right angles and bisect each other. The intersection point divides each diagonal into two equal segments, and also serves as the point of symmetry for the figure.

The diagonals of a rhombus are not only perpendicular but also act as the bisectors of the interior angles. If $E$ denotes the intersection of $AC$ and $BD$, then $AE = EC = \frac{1}{2}AC$ and $BE = ED = \frac{1}{2}BD$, with $AE \perp BE$ at $E$.

General Formula for the Area of a Rhombus Using Base and Height

The area of a rhombus can be expressed as the product of its base and the corresponding altitude (height). Let the length of a side be $a$, and let $h$ denote the perpendicular distance between any pair of parallel sides. The area $A$ is given by

\[ A = a \times h \]

Here, $a$ represents the length of any side, and $h$ represents the vertical height from one side to its parallel counterpart. This formula is a specialisation of the parallelogram area formula, as every rhombus is a parallelogram with all sides equal.

Derivation of the Area Formula in Terms of Diagonals

Let the diagonals $AC$ and $BD$ of rhombus $ABCD$ have lengths $d_1$ and $d_2$ respectively. Let $E$ denote their point of intersection. By the properties of a rhombus, $E$ is the midpoint of both diagonals, and they intersect each other at right angles, i.e., $AC \perp BD$ at $E$.

Each diagonal divides the rhombus into two congruent triangles. Consider $\triangle AEB$, where $A$ and $B$ are adjacent vertices, and $E$ is the midpoint of each diagonal. The lengths $AE = \frac{d_1}{2}$ and $BE = \frac{d_2}{2}$, with $\angle AEB = 90^\circ$.

The area of $\triangle AEB$ can be calculated as follows:

\[ \text{Area of } \triangle AEB = \frac{1}{2} \times AE \times BE \times \sin 90^\circ \]

Since $\sin 90^\circ = 1$, substitute the values of $AE$ and $BE$:

\[ \text{Area of } \triangle AEB = \frac{1}{2} \times \frac{d_1}{2} \times \frac{d_2}{2} \]

Multiplying these factors:

\[ = \frac{1}{2} \times \frac{d_1 d_2}{4} = \frac{d_1 d_2}{8} \]

There are four congruent triangles in the rhombus, so the total area is:

\[ A = 4 \times \frac{d_1 d_2}{8} = \frac{d_1 d_2}{2} \]

Result: The area of a rhombus in terms of its diagonals is $A = \dfrac{d_1 d_2}{2}$.

Area Formula of a Rhombus Using Side and Included Angle

Let the side of the rhombus be $a$ and let $\theta$ represent any interior angle. Since opposite angles are equal in a rhombus, and adjacent angles are supplementary, the area using trigonometry is as follows:

The area of a parallelogram with adjacent sides of length $a$ and $a$, and included angle $\theta$ is

\[ A = a^2 \sin\theta \]

For a rhombus, this formula can use any angle, as the sine function yields the same magnitude for supplementary angles. Both acute and obtuse angles yield positive area due to the periodic nature of the sine function.

Stepwise Derivation: Area in Terms of Side and Angle

Consider consecutive vertices $A$, $B$, and $C$ of $ABCD$ such that $\angle ABC = \theta$. The area is calculated as the product of two consecutive sides and the sine of the included angle:

\[ A = AB \times BC \times \sin\angle B \]

Given $AB = BC = a$, substitute these values:

\[ A = a \times a \times \sin\theta = a^2 \sin\theta \]

Result: For a rhombus with side $a$ and any interior angle $\theta$, $A = a^2 \sin\theta$.

Relationship Between Diagonals and Sides in a Rhombus

From the right triangle formed by half-diagonals and the side of a rhombus, let diagonals be $d_1$ and $d_2$, and side $a$. The points of intersection split the diagonals so that $AE = \frac{d_1}{2}$, $BE = \frac{d_2}{2}$, and the triangle $AEB$ is right-angled at $E$. Applying the Pythagorean theorem yields:

\[ a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]

\[ a^2 = \frac{d_1^2 + d_2^2}{4} \]

\[ 4a^2 = d_1^2 + d_2^2 \]

This formula enables computation of an unknown diagonal or side when the remaining dimensions are known. For dimensional problems where area and one diagonal are known, this relationship permits determination of all relevant parameters.

Worked Example: Area via Base and Height

Given: A rhombus with side $a = 7\,\text{cm}$ and perpendicular height $h = 5\,\text{cm}$.

Substitute values into the formula $A = a \times h$:

\[ A = 7 \times 5 = 35\,\text{cm}^2 \]

Final result: $A = 35\,\text{cm}^2$.

Worked Example: Area via Diagonals

Given: A rhombus with diagonals $d_1 = 12\,\text{cm}$ and $d_2 = 16\,\text{cm}$.

Substitute values into $A = \frac{d_1 d_2}{2}$:

\[ A = \frac{12 \times 16}{2} = \frac{192}{2} = 96\,\text{cm}^2 \]

Final result: $A = 96\,\text{cm}^2$.

Worked Example: Area via Side and Angle

Given: Side $a = 6\,\text{cm}$ and interior angle $\theta = 60^\circ$.

Substitute into $A = a^2 \sin\theta$:

\[ A = 6^2 \times \sin 60^\circ = 36 \times \frac{\sqrt{3}}{2} = 18\sqrt{3}\,\text{cm}^2 \]

Final result: $A = 18\sqrt{3}\,\text{cm}^2$.

Reverse Calculation: Finding a Diagonal Given Area and Other Diagonal

Given: Area $A = 128\,\text{cm}^2$ and one diagonal $d_1 = 16\,\text{cm}$.

The area formula yields $128 = \frac{16 \cdot d_2}{2}$.

Multiply both sides by $2$:

\[ 256 = 16 \cdot d_2 \]

Divide both sides by $16$:

\[ d_2 = \frac{256}{16} = 16\,\text{cm} \]

Final result: The other diagonal $d_2 = 16\,\text{cm}$.

For further study on special cases, reference the Area Of Square Formula and Area Of Hexagon Formula topic pages.

Summary: Key Area Formulas for a Rhombus

The fundamental area expressions for a rhombus are as follows:

1. $A = a \times h$, where $a$ is the length of a side and $h$ the perpendicular height.

2. $A = \dfrac{d_1 d_2}{2}$, where $d_1$ and $d_2$ are the diagonal lengths.

3. $A = a^2 \sin\theta$, where $a$ is the side length and $\theta$ is any interior angle.

For extensions to sectorial or polygonal area computations, see Area Of A Sector Of A Circle Formula and Area Of An Octagon Formula.