7.03 Area of special quadrilaterals

A parallelogram is a quadrilateral with two pairs of opposite sides parallel. A rectangle is a special type of parallelogram but parallelograms do not have to have right angles. Both shapes below are examples of parallelograms.

You may recall that we can find the area of a rectangle using the formula \text{Area}=\text{length}\times\text{width}, and we will see that finding the area of a parallelogram is very similar. We will make use of the base and perpendicular height of the parallelogram to find its area.

Notice that a rectangle is a type of parallelogram, but not all parallelograms are rectangles. Why might this be? Think of what each shape has in common and how they differ.

After using the applet above, we can make the following observations:

• Changing the slant of the parallelogram without changing the base and height did not affect its area. This means that the area of a parallelogram depends only upon its base and its perpendicular height, not the slanted height.

• The base of the parallelogram is the same as the length of the rectangle it creates.

• The perpendicular height of the parallelogram is the same as the width of the rectangle it creates.

• As the area of a rectangle can be found with \text{Area}=\text{length}\times \text{width}, then the area of a parallelogram can be found in a similar way.

Examples

Example 1

Consider the following parallelogram.

a

If the parallelogram is formed into a rectangle, what would the length and width of the rectangle be?

Worked Solution

Create a strategy

The length of the parallelogram is the same as the length of the rectangle. The height of the parallelogram is the width of the rectangle.

Apply the idea

Length: 10\text{ cm}

Width: 8\text{ cm}

Watch question walkthrough

b

Find the area of the parallelogram.

Worked Solution

Create a strategy

Use the area of a parallelogram formula.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle b\times h	Use the area of a parallelogram formula
	\displaystyle =	\displaystyle 10\times8	Substitute b=10 and h=8
	\displaystyle =	\displaystyle 80\text{ m}^2	Evaluate

Watch question walkthrough

Remember that quadrilaterals are polygons with four sides and four vertices. The following are the special quadrilaterals that we will find the area of:

Of course the kite we fly around on a windy day is named after the geometric shape it looks like.

Let's work through how to find the area of each quadrilateral in the following examples.

Examples

Example 2

Find the area of the trapezoid by first calculating the areas of the triangle and rectangle that comprise it.

Worked Solution

Create a strategy

Split the the trapezoid into rectangle and triangle as shown.

Apply the idea

Area of the triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of a triangle formula
\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times(8-6)\times4	Substitute b=(8-6) and h=4
	\displaystyle =	\displaystyle \dfrac12\times2\times4	Evaluate the parentheses
	\displaystyle =	\displaystyle 4	Evaluate

Area of rectangle:

\displaystyle A	\displaystyle =	\displaystyle l \times w	Use the area of rectangle formula
	\displaystyle =	\displaystyle 6\times4	Substitute l=6 and w=4
	\displaystyle =	\displaystyle 24	Evaluate

Total area:

\displaystyle A	\displaystyle =	\displaystyle 4+24	Add the areas
	\displaystyle =	\displaystyle 28\text{ m}^2	Evaluate

Watch question walkthrough

Example 3

The rhombus can be split into two triangles as shown.

Find the area of the rhombus.

Worked Solution

Create a strategy

First find the area of one triangle. To find the height divide the shorter diagonal of the rhombus by 2 because it is divided in half for each triangle. Then find the total area of the rhombus.

Apply the idea

Area of the triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of a triangle formula
\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{2} \times 16 \times(10 \div 2)	Substitute b=16 and h=(10 \div 2)
	\displaystyle =	\displaystyle \dfrac12\times16\times5	Evaluate the parentheses
	\displaystyle =	\displaystyle 40	Evaluate

Total area of the rhombus:

\displaystyle A	\displaystyle =	\displaystyle 2 \times \text{triangle area}	The total is equal to the area of two identical triangles
	\displaystyle =	\displaystyle 2\times40	Substitute 40 for the area of one triangle
	\displaystyle =	\displaystyle 80 \text{ m}^2	Evaluate

Example 4

The kite can be split into two triangles as shown.

a

Find the area of one of the triangles.

Worked Solution

Create a strategy

Use the area of a triangle formula by dividing the shorter diagonal of the kite by 2 for the height of the triangle.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12 \times b \times h	Use the area of a triangle formula
	\displaystyle =	\displaystyle \dfrac12 \times 7 \times ( 4 \div 2)	Substitute b=7 and h=( 4 \div 2)
	\displaystyle =	\displaystyle 7\text{ m}^2	Evaluate

b

Find the area of the kite.

Worked Solution

Create a strategy

Multiply the answer from part (a) by 2.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle 7\times2	Multiply 7 by 2
	\displaystyle =	\displaystyle 14\text{ m}^2	Evaluate