6.6: Area of Rhombus and Kite
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*} and \begin{align*}d_2\end{align*}
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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Date Created:
Feb 23, 2012Last Modified:
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