Area of Kites and Rhombuses
A kite and a rhombus share one area formula, half the product of the diagonals: \(A = \frac{1}{2}xy\). In NSW Year 8 maths (NESA outcome MA4‑ARE‑C‑01) the formula comes from the rectangle drawn around the two diagonals, which the shape fills exactly half of.
You will use only the diagonals and never the sides, double a half-diagonal when that is all the diagram gives, and find a missing diagonal from the area. You will also work on the number plane, count rhombus tiles for a bathroom wall, find the fabric for a kite and a paddock's area in hectares, and take a kite-shaped hole out of a timber panel. It is part of Year 8 Area and Perimeter.
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.