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Area of A Kite – Formulas, Methods & Examples

area of a kite

In geometry, the area of a plane 2D shape is the region covered by the sides of it (region bounded by the perimeter) in a two-dimensional plane. Or we can say that area of any shape is the number of unit squares that can fit into it. Here a unit square refers to a square of side $1$ unit. One good example is graph paper. By counting the number of squares in a region, you can find the area of that region.

Let’s learn about the area of a kite, its formula, and its properties.

area of a kite

Kite – A 2D Plane Figure

A kite is a quadrilateral in which two pairs of adjacent sides are of equal length. No pair of sides in a kite are parallel but one pair of opposite angles are equal. The two diagonals are unequal and intersect at right angles. The longer diagonal of a kite bisects the shorter one.

area of a kite

Properties of A Kite

Following are the characteristic features of a kite

  • A kite has two pairs of adjacent equal sides. Here, $AC = BC$ and $AD = BD$
  • It has one pair of opposite angles (obtuse) that are equal. Here, $\angle A = \angle B$
  • The longer diagonal bisects the shorter one. Here, the diagonal $AB$, $AO = OB$
  • The shorter diagonal forms two isosceles triangles. Here, diagonal $AB$ forms two isosceles triangles: $\triangle ACB$ and $\triangle ADB$. $AC = BC$ and $AD = BD$ in two isosceles triangles
  • The longer diagonal forms two congruent triangles. Here, diagonal $CD$ forms two congruent triangles – $\triangle CAD$ and $\triangle CBD$ by $SSS$ criteria. 
  • The diagonals are perpendicular to each other. Here, $AB \perp CD$
  • The longer diagonal bisects the pair of opposite angles. Here, $\angle ACD = \angle DCB$, and $\angle ADC = \angle CDB$

What is the Area of a Kite?

The area of a kite can be defined as the region covered by the perimeter of a kite in a two-dimensional plane. Like a square, and a rhombus, a kite does not have all four sides equal. The area of a kite is always expressed in terms of $units^{2}$ such as $cm^{2}$, $in^{2}, $m^{2}$, $ft^{2}$ etc. 

Area of a Kite Formula

A kite has two unequal diagonals bisecting each other at right angles. If $d_{1}$ and $d_{2}$ are the lengths of the two diagonals of a kite, its area is given by the formula $A = \frac {1}{2} \times d_{1} \times d_{2}$.

area of a kite
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Derivation of Formula of Area of a Kite Using the Diagonals

Consider a kite $ABCD$, such that the adjacent sides $AB = BC$ and $CD = DA$. The diagonals of the kite are $AC = d_{1}$ and $BD = d_{2}$.

area of a kite

Area of kite $ABCD$ = (Area of $\triangle AOD$) + (Area of $\triangle COD$) + (Area of $\triangle AOB$) + (Area of $\triangle BOC$)

= $\frac {1}{2} \times AO \times DO$ + $\frac {1}{2} \times DO \times CO$ + $\frac {1}{2} \times AO \times BO$ + $\frac {1}{2} \times BO \times CO$

= $\frac {1}{2} \times DO \times \left(AO + CO \right) + \frac {1}{2} \times BO \times \left(AO + CO \right)$

= $\frac {1}{2} \times DO \times AC + \frac {1}{2} \times BO \times AC$

= $\frac {1}{2} \times AC \times (DO + BO) = \frac {1}{2} \times AC \times BD = \frac {1}{2} \times d_{1} \times d_{2}$

Examples

Ex 1: Find the area of a kite whose diagonals are of length $12 cm$ and $8 cm$.

The length of the first diagonal of a kite $d_{1} = 12 cm$

The length of the second diagonal of a kite $d_{2} = 8 cm$

Area of a kite = $\frac {1}{2} \times d_{1} \times d_{2} = \frac {1}{2} \times 12 \times 8 = 48 cm^{2}$

Ex 2: The area of a kite is $120 mm^{2}$. If one of the diagonals is $16 mm$, find the length of the second diagonal.

Area of a kite $A = 120 mm^{2}$

Length of one diagonal of a kite $d_{1} = 16 mm$

Area of a kite $A = \frac {1}{2} \times d_{1} \times d_{2} => 120 = \frac {1}{2} \times 16 \times d_{2} => 120 = 8 \times d_{2} => d_{2} = \frac {120}{8} => d_{2} = 15 mm$

Lowest Form of a Fraction

Conclusion

A kite is a 2D plane figure whose two pairs of adjacent equal sides. The kite has two unequal diagonals intersecting at right angles and you can find the area of a kite by multiplying half by the product of its diagonals.

Practice Problems

  1. Find the area of a kite whose diagonals are of length
    • $14 in$ and $8 in$
    • $11 cm$ and $9 cm$
  2. Find the length of a diagonal, if the area of a kite is $84 cm^{2}$ and the other diagonal is of length $4 cm$.

Recommended Reading

FAQs

How do you find the area of a kite?

The area of a kite can be calculated using the formula Area = $\frac {1}{2} \times d_{1} \times d_{2}$, , where $d_{1}$ and $d_{2}$ are its diagonals.

How do you find the diagonals of a kite?

The length of one diagonal of a kite can be found using the Pythagorean theorem. The length of the other diagonal can be found by substituting the length of the first diagonal into the area of a kite formula if the area is known.

What is the formula for the area of a kite?

The area of a kite is half the product of the lengths of its diagonals. The formula for the area of a kite is given as $\frac {1}{2} \times d_{1} \times d_{2}$, where $d_{1}$ and $d_{2}$ are its diagonals.

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