10.05 Area of parallelograms

A parallelogram is a quadrilateral with two pairs of opposite sides parallel. It looks like a rectangle that has been pushed over.

You may recall that we can find the area of a rectangle using the formula $\text{Area }=\text{length }\times\text{width }$Area =length ×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. Can you work out why? Think of what each shape has in common and how they differ.

 

Parallelograms and rectangles

Parallelograms can be easily rearranged into rectangles. In the applet below, we can rearrange a parallelogram with a base $b$b and a perpendicular height $h$h into a rectangle.

The following guide outlines the key features and concepts in the applet.

1. Click and drag the blue circle on the $\text{Slide }$Slide slider. This will rearrange the parallelogram into a rectangle.
2. Click and drag the blue circle on the $\text{Slant }$Slant slider. Does this change the area of the shape?
3. Click the button $\editable{\text{Change dimensions}}$Change dimensions. You can now adjust the base and height to make a new parallelogram. This can be done with the $b$b slider and $h$h slider or by dragging the vertices of the parallelogram.
4. The area of the parallelogram is being calculated with a formula as you change its dimensions. What part of the calculation changes when you change a dimension? Can you work out the formula?
5. Click the button $\editable{\text{Decompose }}$Decompose to see if the new parallelogram can also rearrange into a rectangle.
6. Click the button $\editable{\text{Show grid}}$Show grid for a grid. Assume that this is a square centimetre grid. To remove it, click the button $\editable{\text{Hide grid}}$Hide grid.

 

Formula for the area of a parallelogram

By 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.
• The perpendicular height of the parallelogram is the same as the width of the rectangle.
• As the area of a rectangle can be found with $\text{Area }=\text{length }\times\text{width }$Area =length ×width , then the area of a parallelogram can be found in a similar way.

Unlike a rectangle, there are generally no right angles in a parallelogram. But we should remember that the height and base are at right angles to each other when we work out the area of a parallelogram.

 

Worked examples

Example 1

Find the area of the parallelogram below.

Think: This parallelogram has a base of $6$6 cm and a height of $4$4 cm. We can rearrange it into a rectangle with length $6$6 cm and width $4$4 cm.

This rectangle has the same area as the parallelogram, which means we can find the area of the parallelogram by calculating the product of its base and height.

Do: We can use the given dimensions in the formula to find the area.

$\text{Area }$Area	$=$=	$\text{base }\times\text{height }$base ×height	(Formula for the area of a parallelogram)
	$=$=	$6\times4$6×4	(Substitute the values for the base and height)
	$=$=	$24$24	(Perform the multiplication to find the area)

So the parallelogram has an area of $24$24 cm².

 

Example 2

What is the area of this parallelogram?

Think: The base always refers to a side of the parallelogram, while the height is the perpendicular distance between two opposite sides. In this parallelogram the base is $12$12 m and the height is $17$17 m.

Do: We can use the given dimensions in the formula to find the area.

$\text{Area }$Area	$=$=	$\text{base }\times\text{height }$base ×height	(Formula for the area of a parallelogram)
	$=$=	$12\times19$12×19	(Substitute the values for the base and height)
	$=$=	$228$228	(Perform the multiplication to find the area)

So the parallelogram has an area of $228$228 m².

Reflect: Sometimes the height will be labelled within the parallelogram, and sometimes it will be convenient to indicate the height with a label outside the parallelogram.

 

Practice questions

Question 1

Question 2

Question 3