Area of a Kite — Formula, Examples & Definition

Area of a Kite

The area of a kite is half the product of the diagonals. Note: This formula works for the area of a rhombus as well, since a rhombus is a special kind of kite. Note that the diagonals of a kite are perpendicular.

Key Formula

A = \frac{1}{2}\,d_1\,d_2

Where:

$A$ = Area of the kite
$d_1$ = Length of the longer diagonal
$d_2$ = Length of the shorter diagonal

Worked Example

Problem: A kite has diagonals of length 10 cm and 6 cm. Find its area.

Step 1: Identify the two diagonals.

d_1 = 10 \text{ cm},\quad d_2 = 6 \text{ cm}

Step 2: Write the area formula for a kite.

A = \frac{1}{2}\,d_1\,d_2

Step 3: Substitute the values and multiply.

A = \frac{1}{2} \times 10 \times 6 = \frac{60}{2}

Step 4: Simplify to get the final area.

A = 30 \text{ cm}^2

Answer: The area of the kite is 30 cm².

Another Example

This example is different from the first because one diagonal is missing. The perpendicular diagonals form right triangles, so the Pythagorean theorem helps find the missing diagonal before using the area formula.

Problem: A kite has side lengths of 10 m and 17 m. Its symmetry diagonal is 21 m. Find the shorter diagonal and the area of the kite.

Step 1: Let the symmetry diagonal be split into two segments, p and q, by the shorter diagonal. Let x be half of the shorter diagonal.

p + q = 21

Step 2: Use the right triangles formed by the perpendicular diagonals. The shorter side of the kite is 10 m and the longer side is 17 m.

p^2 + x^2 = 10^2, \quad q^2 + x^2 = 17^2

Step 3: Subtract the first equation from the second to eliminate x².

q^2 - p^2 = 17^2 - 10^2 = 289 - 100 = 189

Step 4: Factor the difference of squares and use p + q = 21.

(q-p)(q+p)=189, \quad 21(q-p)=189

Step 5: Solve for p and q.

q-p=9, \quad q+p=21 \implies q=15, \ p=6

Step 6: Find x, then double it to get the shorter diagonal.

x^2 = 10^2 - 6^2 = 100 - 36 = 64, \quad x=8, \quad d_2 = 2x = 16 \text{ m}

Step 7: Use the kite area formula with diagonals 21 m and 16 m.

A = \frac{1}{2} \times 21 \times 16 = 168 \text{ m}^2

Answer: The shorter diagonal is 16 m, and the area of the kite is 168 m².

Frequently Asked Questions

Why does the area of a kite formula work?

The two perpendicular diagonals divide the kite into four right triangles. When you pair opposite triangles, they form two rectangles (each with dimensions equal to half of one diagonal and half of the other). The total area of these two rectangles simplifies to ½ d₁ d₂. This reasoning relies on the fact that the diagonals of a kite always intersect at right angles.

Is the area formula for a kite the same as for a rhombus?

Yes. A rhombus is a special case of a kite where all four sides are equal. Because both shapes have perpendicular diagonals, the formula A = ½ d₁ d₂ applies to both. The only difference is that a rhombus's diagonals bisect each other, while in a general kite only one diagonal is bisected by the other.

Does it matter which diagonal is d₁ and which is d₂?

No. Multiplication is commutative, so ½ d₁ d₂ gives the same result regardless of which diagonal you label as d₁ or d₂. The formula works whether you call the longer or shorter diagonal first.

Area of a Kite vs. Area of a Rhombus

	Area of a Kite	Area of a Rhombus
Shape definition	Quadrilateral with two pairs of consecutive congruent sides	Quadrilateral with all four sides congruent
Formula	A = ½ d₁ d₂	A = ½ d₁ d₂
Diagonal properties	Perpendicular; only the symmetry diagonal bisects the other	Perpendicular; both diagonals bisect each other
Relationship	General case	Special case of a kite

Why It Matters

You encounter the area of a kite in geometry courses when studying quadrilaterals, and it frequently appears on standardized tests. The same formula applies to rhombuses, making it a versatile tool. Understanding why perpendicular diagonals lead to this formula also strengthens your grasp of how area formulas are derived from simpler shapes like triangles and rectangles.

Common Mistakes

Mistake: Multiplying the diagonals without dividing by 2.

Correction: The formula is A = ½ d₁ d₂, not d₁ d₂. The product d₁ d₂ gives the area of the rectangle that encloses the kite, which is exactly twice the kite's area. Always remember the factor of one-half.

Mistake: Using the side lengths instead of the diagonal lengths.

Correction: The formula requires the lengths of the two diagonals, not the sides. If you are given side lengths, you need additional information (like one diagonal or the angle between sides) and the Pythagorean theorem to find the diagonals first.

Related Terms

Kite — The shape whose area this formula computes
Area of a Rhombus — Uses the same diagonal-based area formula
Rhombus — A special kite with all sides equal
Diagonal of a Polygon — The two diagonals are the key measurements
Perpendicular — Kite diagonals always meet at right angles
Product — Area is half the product of the diagonals
Formula — General term for the area equation used