NSW Year 7 - 2020 Edition 10.05 Area of parallelograms
Lesson

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

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.
Area of a parallelogram

The area of a parallelogram is given by

$\text{Area }=\text{base }\times\text{height }$Area =base ×height , or

$A=b\times h$A=b×h

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

Complete the table to find the area of the parallelogram shown. 1.  Area $=$= base $\times$× height m2 Area $=$= $\editable{}\times\editable{}$× m2 (Fill in the values for the base and height) Area $=$= $\editable{}$ m2 (Complete the multiplication to find the area)

##### Question 2

Find the area of the parallelogram shown. ##### Question 3

Find the area of a parallelogram whose base is $15$15 cm and height is $7$7 cm.

### Outcomes

#### MA4-13MG

uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area