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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 d_1 and d_2

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

These are the same as the sides of the _________________________.

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

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 d_1d_2 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 d_1 and d_2

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

These are the same as the sides of the _________________________.

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

The area of the rectangle is d_1d_2 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.

{\;}

{\;}

{\;}

{\;}

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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Authors:

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Grades:

8 , 9 , 10

Date Created:

Feb 23, 2012

Last Modified:

May 12, 2014
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