# 6.6: Area of Rhombus and Kite

**At Grade**Created by: CK-12

## Learning Objectives

- Use formulas to find the area of rhombuses and kites – quadrilaterals with perpendicular diagonals.

## Area of a Rhombus or Kite

First let’s start with a review of some of the properties of a **kite** and a **rhombus**.

A reminder: the **diagonals** of both a kite and a rhombus are the dotted lines in the figures above. **Diagonals** connect opposite vertices.

Does it have...? |
Kite |
Rhombus |
---|---|---|

Congruent sides | Yes, 2 pairs | Yes, all 4 |

Opposite angles congruent | 1 pair yes. 1 pair maybe | Both pairs yes |

Perpendicular diagonals | Yes | Yes |

Diagonals bisected | 1 yes. 1 maybe | Both yes |

Now you are ready to develop area formulas. For both a bite and a rhombus, we will “Frame it in a rectangle.” Here’s how you can frame a **rhombus** in a rectangle:

Notice that:

- The
*base*and*height*of the**rectangle**are the same as the lengths of the two*diagonals*of the**rhombus**. - The rectangle is divided into 8 congruent triangles; 4 of the triangles fill the rhombus, so the
**area**of the rhombus is*half of the area of the rectangle*.

**Area of a Rhombus with Diagonals \begin{align*}d_1\end{align*} d1 and \begin{align*}d_2\end{align*}d2**

The diagonals of the rhombus in the figure below are labeled ______ and ______.

These are the same as the sides of the _________________________.

\begin{align*}A = \frac{1}{2} d_1 d_2 = \frac{d_1 d_2}{2}\end{align*}

As you learned on the previous page: The area of a **rhombus** is half of the area of the ___________________ it is framed in.

The area of the rectangle is \begin{align*}d_1d_2\end{align*} so half the area of the rectangle is _______________.

Therefore, the **area** of the **rhombus** is __________________.

Next, we will examine the kite. We will follow the same rule: “Frame it in a rectangle.” Here’s how you can frame a **kite** in a rectangle:

Notice that:

- The
*base*and*height*of the**rectangle**are the same as the lengths of the two*diagonals*of the**kite**. - The rectangle is divided into 8 triangles; 4 of the triangles fill the kite. For every triangle inside the kite, there is a
*congruent*triangle outside the kite. So, the**area**of the kite is*half of the area of the rectangle*.

Just like a rhombus:

The area of a **kite** is half of the area of the ___________________ it is framed in.

**Area of a Kite with Diagonals \begin{align*}d_1\end{align*} and \begin{align*}d_2\end{align*}**

The diagonals of the kite in the figure below are labeled ______ and ______.

These are the same as the sides of the _________________________.

\begin{align*}A = \frac{1}{2} d_1 d_2 = \frac{d_1 d_2}{2}\end{align*}

The area of the rectangle is \begin{align*}d_1d_2\end{align*} so half the area of the rectangle is _______________.

Therefore, the **area** of the **kite** is __________________.

**Reading Check**

1. *Fill in the blank:*

*A kite is a quadrilateral with _____________________________ diagonals*.

2. *Fill in the blank:*

*A rhombus is a quadrilateral with perpendicular* __________________________.

3. *What is a diagonal? Describe it in your own words*.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

4. *True or false: All 4 sides of a rhombus are congruent*.

5. *True or false: No interior angles in a kite are congruent*.

6. *True or false: When you frame either a kite or a rhombus in a rectangle, the diagonals of the kite or rhombus are the* *same**as the base and height of the rectangle*.

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