Area of a Rhombus

The area of a rhombus is equal to half the product of its diagonals: $$ A = \frac{d_1 \cdot d_2}{2} $$

This is one of the most useful formulas for working with rhombuses because it allows you to calculate the area directly from the lengths of the diagonals, without needing to know the height or any angles.

Inverse Formulas

If you know the area of a rhombus and the length of one diagonal, you can easily find the other diagonal.

Starting from the area formula:

$$ A = \frac{d_1 \cdot d_2}{2} $$

Solving for each diagonal gives:

$$ d_1 = \frac{2A}{d_2} $$

$$ d_2 = \frac{2A}{d_1} $$

These formulas are particularly useful when solving geometry problems where some measurements are missing.

A Practical Example

Consider a rhombus whose diagonals measure d₁ = 3 cm and d₂ = 4 cm.

Using the area formula:

$$ A = \frac{d_1 \cdot d_2}{2} $$

Substituting the known values:

$$ A = \frac{3 \ cm \cdot 4 \ cm}{2} $$

$$ A = \frac{12 \ cm^2}{2} $$

$$ A = 6 \ cm^2 $$

Therefore, the area of the rhombus is 6 cm².

Proof

Let's see why the formula works.

Consider a rhombus ABCD.

Draw lines through the vertices that are perpendicular to the diagonals.

These lines form the rectangle EFGI.

The side lengths of the rectangle are equal to the lengths of the diagonals of the rhombus:

$$ d_1 = \overline{AC} = \overline{EF} = \overline{GI} $$

$$ d_2 = \overline{BD} = \overline{EI} = \overline{FG} $$

The rectangle is divided into eight congruent triangles, while the rhombus is divided into four congruent triangles.

As a result, the rhombus occupies exactly half the area of the rectangle.

The area of the rectangle is:

$$ A_{rectangle} = b \cdot h $$

Therefore, the area of the rhombus is:

$$ A = \frac{b \cdot h}{2} $$

Since the rectangle's dimensions are equal to the diagonals of the rhombus, with b = d₁ and h = d₂, we obtain:

$$ A = \frac{d_1 \cdot d_2}{2} $$

This proves the formula for the area of a rhombus.

Observations

The area formula for a rhombus leads to several useful geometric relationships.

• A rhombus is a special type of parallelogram with four equal sides. Therefore, its area can also be calculated using the standard parallelogram formula: $$ A = b \cdot h $$ where b is the base and h is the corresponding height.
• A square is a special case of a rhombus whose diagonals are equal in length. In this case: $$ A = \frac{d \cdot d}{2} = \frac{d^2}{2} $$ Rearranging the formula gives: $$ d^2 = 2A $$ Taking the square root of both sides: $$ d = \sqrt{2A} $$

These relationships make it easier to solve a wide range of geometry problems involving rhombuses, squares, and other quadrilaterals.