The area of a kite equals half the product of its two diagonals: A = ½ × d₁ × d₂. This single formula works for any kite regardless of its proportions, because the diagonals of a kite always cross at right angles with one bisecting the other.
What is a kite?
A kite is a quadrilateral with two pairs of adjacent equal sides. It looks like the flying toy it is named after.
Key properties of a kite:
- Two pairs of equal adjacent sides (not opposite sides like a parallelogram).
- One diagonal (the "axis of symmetry") bisects the other diagonal at right angles.
- The axis of symmetry bisects the pair of angles at the "tips" of the kite.
- One pair of opposite angles are equal (the angles between the unequal sides).
Example dimensions: a kite with sides of 5 cm and 8 cm, where the longer diagonal (axis of symmetry) is 10 cm and the shorter diagonal is 6 cm.
What is the formula for the area of a kite?
Area = ½ × d₁ × d₂
where d₁ and d₂ are the full lengths of the two diagonals.
Worked example 1: A kite has diagonals of length 8 cm and 5 cm.
Area = ½ × 8 × 5 = 20 cm²
Worked example 2: A kite-shaped garden has diagonals of 12 m and 7 m.
Area = ½ × 12 × 7 = 42 m²
Why does the formula work? (The proof)
Label the kite so that the axis of symmetry d₁ runs vertically and is divided by the perpendicular diagonal d₂. Let d₁ be split into a top portion p and a bottom portion q (where p + q = d₁), and d₂ be bisected into two halves d₂/2 on each side.
The kite is divided into four right-angled triangles:
- Two triangles with legs p and d₂/2
- Two triangles with legs q and d₂/2
Area of all four triangles:
= 2 × (½ × p × d₂/2) + 2 × (½ × q × d₂/2)
= (p × d₂/2) + (q × d₂/2)
= (d₂/2)(p + q)
= (d₂/2) × d₁
= ½ × d₁ × d₂ ✓
This proof holds regardless of where the crossing point lies on d₁, which is why the formula works for every kite.
How do you find a diagonal if the area is given?
Rearrange the formula: d₁ = 2A / d₂ (or d₂ = 2A / d₁).
Worked example: A kite has area 36 cm² and one diagonal of length 9 cm. Find the other diagonal.
d₂ = (2 × 36) / 9 = 72 / 9 = 8 cm
Does the same formula apply to a rhombus?
Yes — a rhombus is a special kite where all four sides are equal and both diagonals bisect each other. The diagonals still cross at right angles, so:
Area of rhombus = ½ × d₁ × d₂
Example: A rhombus has diagonals of 10 cm and 6 cm. Area = ½ × 10 × 6 = 30 cm²
| Shape | Formula for area | Special condition |
|---|---|---|
| Kite | ½ × d₁ × d₂ | One diagonal bisects the other at 90° |
| Rhombus | ½ × d₁ × d₂ | Both diagonals bisect each other at 90° |
| Square | ½ × d × d = ½d² | Both diagonals equal and bisect at 90° |
What mistakes should you avoid?
Mistake 1 — Using the sides instead of the diagonals. The formula requires the diagonals d₁ and d₂, not the side lengths. Read the question carefully to identify which measurements are diagonals.
Mistake 2 — Forgetting the ½. The product d₁ × d₂ without the ½ gives twice the area. The ½ is essential.
Mistake 3 — Confusing the kite formula with the triangle formula. The area of a triangle is ½ × base × height. The kite formula is ½ × d₁ × d₂ (two diagonals). Although they look similar, the base and height in a triangle are perpendicular sides, while d₁ and d₂ in a kite are the full diagonal lengths.
Frequently asked questions
Do the diagonals of a kite always cross at right angles?
Yes — this is a defining property of a kite. The axis of symmetry (the longer diagonal) is a line of symmetry, and the shorter diagonal must cross it perpendicularly for the two halves to be mirror images. It is this 90° angle that makes the diagonal formula work.
Can a kite be a square or a rectangle?
A square is a special case of a rhombus, and a rhombus is a special case of a kite (where all four sides happen to be equal). So in the broader mathematical hierarchy, a square is a kite — and the diagonal formula gives the correct area for it. A rectangle is not a kite because its sides come in pairs of equal opposite sides, not adjacent equal sides.
What is the perimeter of a kite?
A kite has two pairs of adjacent equal sides. If the short sides have length a and the long sides have length b, the perimeter is 2a + 2b. This is different from the area formula (which uses diagonals), so make sure you use the side lengths for perimeter.
How do I find the area of a kite from its sides if no diagonal is given?
This is more complex and requires trigonometry (not typically asked at KS3). If you know all four sides and one angle, you can find the diagonal using the cosine rule, then apply the area formula. At KS3, questions about kite area will always supply both diagonal lengths.
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