Area of a Rhombus or Kite

Statement

The area of a rhombus or a kite can be calculated using the lengths of its diagonals. The formula is:

\[ A = \tfrac{1}{2} d_1 d_2 \]

Here, \(d_1\) and \(d_2\) are the lengths of the two diagonals that intersect at right angles.

Why it’s true (short reason)

• A rhombus or kite can be split into four right-angled triangles by its diagonals.
• The diagonals intersect at right angles, so each triangle has area \(\tfrac{1}{2}\times \tfrac{d_1}{2}\times \tfrac{d_2}{2}\).
• Adding the four triangles gives \(A = \tfrac{1}{2} d_1 d_2\).

Recipe (how to use it)

1. Measure the two diagonals, \(d_1\) and \(d_2\).
2. Multiply them together.
3. Halve the result: \[ A = \tfrac{1}{2} d_1 d_2. \]
4. Write the answer with correct square units (cm², m², etc.).

Spotting it

Look for this formula when:

• The shape is clearly a rhombus or kite.
• The lengths of both diagonals are provided or can be calculated.
• The diagonals are perpendicular (this property is guaranteed for rhombus/kite).

Common pairings

• Pythagoras’ theorem, to find diagonal lengths when sides are given.
• Trigonometry, for cases where diagonals must be worked out indirectly.
• Comparisons with parallelogram or triangle area formulas.

Mini examples

1. Given: A rhombus with diagonals 12 cm and 16 cm. Find: Area. Answer: \(A = \tfrac{1}{2}\times 12 \times 16 = 96\ \text{cm}^2\).
2. Given: A kite with diagonals 9 m and 14 m. Find: Area. Answer: \(A = \tfrac{1}{2}\times 9 \times 14 = 63\ \text{m}^2\).

Pitfalls

• Confusing the formula with rectangle area \(A = lw\). Here, we must use diagonals.
• Forgetting to halve the product of diagonals.
• Mixing up units, e.g. one diagonal in cm and the other in mm.
• Applying the formula to non-rhombus quadrilaterals where diagonals are not perpendicular (not valid).

Exam strategy

• Check shape type: only use for rhombus or kite (perpendicular diagonals).
• Underline diagonal values to avoid confusing them with side lengths.
• Multiply carefully and halve last — students often forget the 1/2 factor.
• Convert units first if necessary (e.g. all into cm before calculation).

Summary

The formula \(A = \tfrac{1}{2} d_1 d_2\) offers a quick and accurate way to calculate the area of rhombuses and kites by using their diagonals. It works because the diagonals split the quadrilateral into four right-angled triangles. In exams, look for the diagonals, check they are perpendicular, and apply the formula directly. Always halve the product of the diagonals and present your answer with correct units.