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Kite — Definition, Formula & Properties

Kite

A quadrilateral with two pairs of adjacent sides that are congruent. Note that the diagonals of a kite are perpendicular.

 

Kite

d1 = long diagonal of kite
d2 = short diagonal of kite

Area = (½) d1d2


Kite shape with perpendicular diagonals labeled d₁ (long, vertical) and d₂ (short, horizontal), intersecting at a right angle.

 

See also

Rhombus, square, area of a kite

Key Formula

A=12 d1⋅d2A = \frac{1}{2}\,d_1 \cdot d_2
Where:
  • AA = Area of the kite
  • d1d_1 = Length of the longer diagonal
  • d2d_2 = Length of the shorter diagonal

Worked Example

Problem: A kite has diagonals of length 10 cm and 6 cm. Find the area of the kite.
Step 1: Identify the two diagonals. Here, the longer diagonal is 10 cm and the shorter diagonal is 6 cm.
d1=10 cm,d2=6 cmd_1 = 10 \text{ cm}, \quad d_2 = 6 \text{ cm}
Step 2: Write the area formula for a kite.
A=12 d1⋅d2A = \frac{1}{2}\,d_1 \cdot d_2
Step 3: Substitute the values into the formula.
A=12(10)(6)A = \frac{1}{2}(10)(6)
Step 4: Multiply to get the area.
A=12(60)=30 cm2A = \frac{1}{2}(60) = 30 \text{ cm}^2
Answer: The area of the kite is 30 cm².

Another Example

This example shows how to find the missing diagonal using side lengths and the Pythagorean theorem, since the kite's diagonals are perpendicular. It illustrates a common exam scenario where only one diagonal and the side lengths are given.

Problem: A kite has two pairs of adjacent congruent sides: one pair is 5 cm each and the other pair is 12 cm each. The longer diagonal measures 16 cm. Find the area of the kite.
Step 1: Label the kite ABCD so that AB = AD = 12 cm and CB = CD = 5 cm. The longer diagonal AC connects the vertices between the unequal pairs of sides. The shorter diagonal BD crosses AC at right angles at point E.
Step 2: The longer diagonal splits the kite into two congruent triangles. Because the diagonals are perpendicular, triangle ABE is a right triangle. The longer diagonal AC = 16 cm. In a kite, the shorter diagonal is bisected by the longer diagonal, so BE = ED. We need to find BE. First, find AE using triangle ABE.
Step 3: Let AE = x, so EC = 16 − x. In right triangle ABE: AB² = AE² + BE², giving 12² = x² + BE². In right triangle CBE: CB² = CE² + BE², giving 5² = (16 − x)² + BE².
144=x2+BE2and25=(16−x)2+BE2144 = x^2 + BE^2 \qquad \text{and} \qquad 25 = (16 - x)^2 + BE^2
Step 4: Subtract the second equation from the first to eliminate BE².
144−25=x2−(16−x)2144 - 25 = x^2 - (16 - x)^2
Step 5: Expand and solve for x. The right side becomes x² − 256 + 32x − x² = 32x − 256. So 119 = 32x − 256, which gives 32x = 375 and x = 375/32. Now find BE² = 144 − (375/32)² = 144 − 140625/1024 = (147456 − 140625)/1024 = 6831/1024. Then BE = √6831 / 32, and the full shorter diagonal is d₂ = 2·BE = 2√6831 / 32 = √6831 / 16. Rather than continue with this messy value, use the area formula directly: A = ½ · d₁ · d₂ = ½ · 16 · (√6831/16) · 2. Instead, it is simpler to use the two triangles directly.
Step 5 (simplified): A cleaner approach: the area of the kite equals the sum of two triangles sharing diagonal AC. Triangle ABC has base AC = 16 and height BE. Triangle ACD has base AC = 16 and height DE = BE. So A = 2 × (½ · 16 · BE) = 16 · BE. From Step 3, BE² = 144 − (375/32)² = 6831/1024, so BE ≈ 2.583 cm. Area ≈ 16 × 2.583 ≈ 41.3 cm². Alternatively, just apply the diagonal formula once you know both diagonals. The short diagonal d₂ = 2(2.583) ≈ 5.17 cm, so A ≈ ½(16)(5.17) ≈ 41.3 cm².
A=12(16)(5.17)≈41.3 cm2A = \frac{1}{2}(16)(5.17) \approx 41.3 \text{ cm}^2
Answer: The area of the kite is approximately 41.3 cm².

Frequently Asked Questions

What is the difference between a kite and a rhombus?
A kite has two distinct pairs of adjacent congruent sides, whereas a rhombus has all four sides congruent. Every rhombus is technically a special case of a kite (where the two pairs happen to be equal), but most kites are not rhombuses. Both shapes have perpendicular diagonals, so they share the same area formula: A = ½ d₁ d₂.
Are the diagonals of a kite always perpendicular?
Yes. One of the defining properties of a kite is that its diagonals always intersect at right angles (90°). Additionally, the longer diagonal bisects the shorter diagonal, but the shorter diagonal does not necessarily bisect the longer one.
Can a kite have two right angles?
Yes. In many kites, the two angles where the unequal sides meet are not right angles, but a kite can have exactly two right angles. These right angles occur at the endpoints of the shorter diagonal. A square is a special kite that has four right angles.

Kite vs. Rhombus

KiteRhombus
DefinitionQuadrilateral with two pairs of adjacent congruent sides (pairs can differ in length)Quadrilateral with all four sides congruent
DiagonalsPerpendicular; longer diagonal bisects the shorter onePerpendicular; each diagonal bisects the other
Area formulaA = ½ d₁ d₂A = ½ d₁ d₂
SymmetryOne line of symmetry (along the longer diagonal)Two lines of symmetry (along each diagonal)
RelationshipA rhombus is a special type of kiteA rhombus is also a parallelogram; a kite generally is not

Why It Matters

Kites appear frequently in geometry courses when studying properties of quadrilaterals, diagonal relationships, and area formulas. Standardized tests often give you the diagonals of a kite and ask for the area, so knowing the formula A = ½ d₁ d₂ is essential. The perpendicular-diagonal property also makes kites a useful context for applying the Pythagorean theorem.

Common Mistakes

Mistake: Confusing adjacent congruent sides with opposite congruent sides.
Correction: In a kite, the two congruent sides in each pair are next to each other (adjacent), not across from each other. A quadrilateral with opposite sides congruent is a parallelogram, not a kite.
Mistake: Assuming both diagonals are bisected by the other.
Correction: Only the shorter diagonal is bisected by the longer one. The longer diagonal is not bisected by the shorter diagonal unless the kite is also a rhombus.

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