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As a matter of fact, there is! In this article, we will discuss a formula that calculates the area of kites and observe several worked examples that employ this technique.

## Recap. Defining a Kite

Before we begin, let us begin by refreshing our memories of kites. A **kite **is a type of quadrilateral that has two pairs of equal adjacent sides. Like all other quadrilaterals, it contains 4 sides, 4 angles, and 2 diagonals.

The structure of a kite satisfies the characteristics of a **cyclic quadrilateral**. A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a circle. It is sometimes referred to as an inscribed quadrilateral. The circle that holds all four of these vertices on its circumference is called the circumcircle or circumscribed circle. Here is a diagram of a kite within a circle.

Example of a cyclic quadrilateral

### Properties of a Kite

Let us now recall the fundamental properties of a kite. Here we have a kite denoted by ABCD. M is the point at which the diagonals intersect.

Diagram of a kite

The following table is a list of its features.

Properties of a Kite | Description |

It has two pairs of equal adjacent sides | AB = BC and AD = DC |

It has one pair of equal opposite angles that are obtuse | ∠BAD = ∠BCD > 90 |

No parallel lines | |

It has two non-equal diagonals | AC ≠ BD |

Diagonals are perpendicular and bisect each other | AC ⊥ BD and AM = MC and BM = MD |

We are now ready to learn more about the area of a Kite.

**The Area Formula for a Kite **

The area of a kite is the space bounded by its sides. Referring back to our previous diagram of a kite, the area formula is given by

\[A=\frac{1}{2}\times d_1 \times d_2\]

where \(d_1\) and \(d_2\) are the lengths of the vertical diagonal and horizontal diagonal, respectively.

Area of a kite

### Deriving the Area of a Kite

Now we have an explicit recipe for finding the area of a kite. But how did it come about? This segment will discuss a step of step derivation of how this formula actually satisfies the area of a given kite. Again, let's turn our attention back to our previous kite, shown below.

Area of a kite

For our kite ABCD above, let's call the length of the shorter diagonal \(AC=x\) and the length of the longer diagonal \(BD=y\). From the properties of a kite, both these diagonals are perpendicular (at right angles) and bisect each other.

With that in mind, we have

\[AM=MC=\frac{AC}{2}=\frac{x}{2}\]

The area of kite ABCD is made up of the sum of two areas: triangle ABD and triangle BCD. Writing this as an expression, we have

Area of kite ABCD = Area of ΔABD + Area of ΔBCD

Let's called this Equation 1.

The area of a triangle is the product of its base and height multiplied by half, that is,

\[\text{Area of a Triangle}=\frac{1}{2}\times b \times h\]

where \(b\) is the base and \(h\) is the height. Using this formula, let's determine the areas of triangle ABD and triangle BCD.

\[\text{Area of Triangle ABD}=\frac{1}{2}\times AM\times BD\]

\[\text{Area of Triangle BCD}=\frac{1}{2}\times MC\times BD\]

Now replacing AM, BD and MC by \(x\) and \(y\), we have

\[\text{Area of Triangle ABD}=\frac{1}{2}\times\frac{x}{2}\times y=\frac{xy}{4}\]

\[\text{Area of Triangle BCD}=\frac{1}{2}\times\frac{x}{2}\times y=\frac{xy}{4}\]

Now, using Equation 1, we obtain

\[\text{Area of kite ABCD}=\frac{xy}{4}+\frac{xy}{4}=\frac{xy}{2}\]

Finally, substituting the values of \(x\) and \(y\), we have the required formula for the area of a kite.

\[\text{Area of a kite}=\frac{1}{2}\times AC \times BD\]

## The Area of a Kite and a Rhombus

The area formula of a kite happens to follow the same idea as the area of a rhombus. Let's recall the structure of a rhombus. Here we have a rhombus denoted by ABCD. M is the point at which the diagonals intersect.

Diagram of a rhombus

You can already see the resemblances with a kite, just by looking at this diagram. The following table is a list of its features.

Properties of a Rhombus | Description |

Has four equal sides | AB = BC = CD = DA |

Has opposite angles of equal measures | ∠ABC = ∠CDA and ∠BCD = ∠DAB |

Has two pairs of parallel sides | AB // DC and AD // BC |

Has two non-equal diagonals | AC ≠ BD |

Diagonals are perpendicular and bisect each other | AC ⊥ BD and AM = MC and BM = MD |

**The Area Formula for a Rhombus**

\[A=\frac{1}{2}\times d_1 \times d_2\]

where d_{1} and d_{2} are the lengths of the vertical diagonal and horizontal diagonal, respectively.

Area of a rhombus

## Examples of Area of Kites

In this section, we shall look at several worked examples that make use of this formula that deduces the area of a kite. Here is the first example.

Cathy has 3 identical kite-shaped notecards with diagonals of lengths 5 inches and 17 inches. Determine the sum of the area for these 3 notecards.

**Solution**

The diagonals of each box are given by \(d_1=5\) and \(d_2=17\). Using the area formula for a kite, the area of one notecard is

\[A=\frac{1}{2}\times 5 \times 17=\frac{82}{2}=42.5\]

Thus, the area of each kite is 42.5 in^{2}. Since we have 3 identical notecards, we can simply multiply this area by 3 to find their total area.

\[42.5\times 3=127.5\]

Thus, the total area of all 3 notecards is 127.5 in^{2}.

Let's look at another example.

Mary has a cardboard cutout shaped like a kite. The shorter diagonal measures 3 feet while the longer diagonal measured is 14 feet. What is the area of this cutout?

She then decides to divide this cutout into 7 separate pieces of equal areas. What would the area of each piece be?

**Solution **

The diagonals of this cutout are given by \(d_1=3\) and \(d_2=14\). Using the area formula for a kite, the area of this cutout is

\[A=\frac{1}{2}\times 3 \times 14=\frac{42}{2}=21\]

Thus, the area of this cutout is 21 ft^{2}. Since Mary wants to divide this cutout into 7 identical segments, we can simply divide this area by 7 to identify the area of each piece.

\[\frac{21}{7}=3\]

Hence, the area of each piece would be 3 ft^{2}.

Here is one last example before we end this topic.

David has a kite with an area of 304 square inches. The shorter diagonal is 16 inches long. What is the length of the longer diagonal?

**Solution**

In this question, we are given the measures of the area and one of the diagonals of this kite, namely \(A=304\) and \(d_1=16\). In order to find the length of the longer diagonal, \(d_2\), we need to rearrange the given formula to make \(d_2\) the subject. Given that the formula for the area of a kite is

\[A=\frac{1}{2}\times d_1 \times d_2\]

Rearranging this formula so that \(d_2\) becomes the subject yields

\[d_2=\frac{2A}{d_1}\]

Now substituting our known values for \(A\) and \(d_1\), we have

\[d_2=\frac{2\times 304}{16}=38\]

Thus, the length of the longer diagonal is 38 inches.

## Area of Kites - Key takeaways

- A kite is a type of quadrilateral with no parallel lines.
- A kite has two pairs of equal adjacent sides and one pair of equal opposite angles that are obtuse.
- A kite has two non-equal diagonals.
- The diagonals of a kite are perpendicular and bisect each other.
- The area of a kite is given by \[A=\frac{1}{2}\times d_1 \times d_2\] where \(d_1\) and \(d_2\) are the lengths of the vertical diagonal and horizontal diagonal, respectively.

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##### Frequently Asked Questions about Area of a Kite

How to find the area of a kite?

The area of a kite is the product of both its diagonals multiplied by half.

What is the area of a kite?

The area of a kite is the product of both its diagonals multiplied by half.

What are the properties of a kite?

- A kite is a type of quadrilateral.
- A kite has two pairs of equal adjacent sides.
- A kite has one pair of equal opposite angles that are obtuse.
- A kite has no parallel lines.
- A kite has two non-equal diagonals.
- The diagonals of a kite are perpendicular and bisect each other.

Are diagonals of a kite congruent?

No

Are the diagonals of a kite perpendicular?

Yes

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