Area of Kites and Rhombuses
Theory
The area of a kite or a rhombus is half the product of its diagonals: \(A = \dfrac{1}{2}xy\), where \(x\) and \(y\) are the full diagonals. A kite with diagonals \(6\) cm and \(9\) cm has area \(\dfrac{1}{2} \times 6 \times 9 = 27~\text{cm}^2\). This Year 8 maths page shows how.
A diagonal joins two opposite corners. For a kite or a rhombus with diagonals \(x\) and \(y\): \(A = \dfrac{1}{2}xy\). The side lengths are never used.
Why it works. The shape fills exactly half of the rectangle built on its diagonals.
The diagonals cross at right angles. In a rhombus each diagonal cuts the other in half. In a kite the axis of symmetry cuts the other diagonal in half, but is not cut in half itself. A square is a special rhombus.
Given only a half? In a rhombus every centre-to-corner length is half a diagonal, so double it. In a kite only the diagonal cut by the axis has equal halves.
Missing diagonal: substitute, double both sides to remove the half, then divide. If one diagonal is a multiple of the other, call the shorter one \(x\); you get an \(x^2\), so square root.
Units. Change both diagonals to one unit first; \(1\) ha \(= 10\,000~\text{m}^2\). To count tiles, divide the total area by one tile’s area.
Kites and rhombuses:
How to find the area
- Find the full lengths of both diagonals, in one unit (double any halves).
- Multiply them and halve.
- Give square units, then answer the question asked.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 6 \times 9 = 27\) |
\(27~\text{cm}^2\)
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 8 \times 14 = 56\) |
\(56~\text{cm}^2\)
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 16 \times 11\) |
| \(\) | \(=\) | \(8 \times 11 = 88\) |
\(88~\text{cm}^2\)
The axis cuts the other diagonal in half, so double the \(5\) cm: \(10\) cm.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 10 \times 18 = 90\) |
\(90~\text{cm}^2\)
| \(2.4\) | \(=\) | \(\dfrac{1}{2} \times 1.6 \times y\) |
| \(2.4\) | \(=\) | \(0.8y\) |
| \(y\) | \(=\) | \(3\) |
\(160\) cm \(= 1.6\) m, so the other diagonal is \(3\) m. Check: \(\dfrac{1}{2} \times 1.6 \times 3 = 2.4\).
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 240 \times 150\) |
| \(\) | \(=\) | \(18\,000~\text{m}^2\) |
| \(18\,000 \div 10\,000\) | \(=\) | \(1.8\) |
\(1.8\) ha
| \(\dfrac{1}{2} \times x \times 4x\) | \(=\) | \(50\) |
| \(2x^2\) | \(=\) | \(50\) |
| \(x^2\) | \(=\) | \(25\) |
| \(x\) | \(=\) | \(5\) |
A length is positive, so \(x = 5\): the diagonals are \(5\) cm and \(20\) cm.
Common pitfalls
Frequently asked questions
What is the area of a rhombus?
Half the product of its diagonals: \(A = \dfrac{1}{2}xy\).
What is the area of a kite?
Also \(A = \dfrac{1}{2}xy\), using its two diagonals.
Why does the formula have a half?
The shape fills exactly half of the rectangle built on its diagonals.
How do I find a missing diagonal?
Double the area, then divide by the known diagonal: \(y = 2A \div x\).
Can I use this formula for a square?
Yes. A square is a special rhombus, so half the product of its equal diagonals gives its area.