Middle Years

# 9.01 Areas of special quadrilaterals

Lesson

## Rectangles and squares

Area is a measure of the space inside a two-dimensional shape. We have previously looked at how to find the area of a rectangle or square, using the following rule:

Area of a rectangle or square

$\text{Area of a rectangle}=\text{length }\times\text{width }$Area of a rectangle=length ×width  or $A=lw$A=lw

$\text{Area of a square}=\text{side }\times\text{side }$Area of a square=side ×side  or $A=s^2$A=s2

But how can we find the area of other quadrilaterals, such as a rhombuses, parallelograms, trapeziums and kites?

## Parallelograms

A parallelogram is a four-sided shape where the opposite sides are parallel. It is similar to a rectangle, but doesn't necessarily have right angles. This means a rectangle is a special type of parallelogram.

The good news is, we can use the similarities between parallelograms and rectangles to calculate the area of a parallelogram. The following applet explores these similarities further.

Because a parallelogram might not have right angles, we need to make sure we use the perpendicular height when we work out the area. It doesn't matter which side we use as the base, as long as the height is the perpendicular height with respect to the base.

Area of a parallelogram
$\text{Area of a parallelogram}=\text{base }\times\text{height }$Area of a parallelogram=base ×height  or $A=bh$A=bh

#### Practice questions

##### Question 1

If this parallelogram is cut along the dotted line, the pieces can be rearranged to form a rectangle:

1. Complete the table to find the area of the rectangle.

 $\text{Area of rectangle }$Area of rectangle $=$= $\text{length }\times\text{width }$length ×width cm2 $A$A $=$= $\editable{}\times\editable{}$× cm2 (Fill in the values for the length and width) $A$A $=$= $\editable{}$ cm2 (Complete the multiplication to find the area)
2. Now find the area of the parallelogram.

##### Question 2

By rearranging the parallelogram into a rectangle, find its area.

## Rhombuses

A rhombus is a four-sided shape with four equal sides and opposite sides parallel. It is similar to a square, but doesn't necessarily have right angles. This means a square is a special type of rhombus, and a rhombus is a special type of parallelogram.

If we are given the base and perpendicular height, then we can just use the formula for the area of a parallelogram. However, we are usually given the lengths of the diagonals, so we will need another method for finding the area of a rhombus.

Once again, we can use the similarities between rhombuses and rectangles to find their area. The following applet explores these similarities further.

By turning a rhombus into a rectangle, we can see that we end up with a rectangle that has one dimension equal to the length of one of the diagonals while the other dimension will be half the other diagonal. Notice that it does not matter which diagonal we half.

Area of a rhombus
$\text{Area of a rhombus }=\frac{1}{2}\times\text{diagonal }x\times\text{diagonal }y$Area of a rhombus =12×diagonal x×diagonal y or $A=\frac{1}{2}xy$A=12xy

#### Practice questions

##### Question 3

The rhombus on the left can be split into two triangles as shown.

1. Find the area of one of the triangles.

2. Now find the area of the rhombus.

##### Question 4

Find the area of the rhombus shown.

## Trapeziums

Trapeziums (or "trapezia") are related to parallelograms. That's important because it means we can use the area of a parallelogram to help us work out the area of a trapezium.

The following applets explores how a trapezium is exactly half of its related parallelogram.

We could break it up into a rectangle, and either one or two triangles, depending on its starting shape, and calculate each of these shapes individually.

We can also turn a trapezium into a rectangle, as demonstrated in the image below. By cutting it in the appropriate position, we can rearrange it to form a rectangle. The length $l$l will be the average length of the two parallel sides, $6$6 and $13$13, so $9.5$9.5 will be the length of the rectangle formed.

A trapezium transformed into a rectangle with dimensions $8$8 cm by $9.5$9.5 cm

Area of a trapezium
$\text{Area }=\frac{1}{2}\times\left(\text{side a}+\text{side b}\right)\times\text{height h}$Area =12×(side a+side b)×height h
The sides $a$a and $b$b are the lengths of the parallel sides, and $h$h is the perpendicular height.

#### Practice questions

##### Question 5

Consider the trapezium shown below which has been split into a rectangle and a right-angled triangle.

1. Find the area of the rectangle.

2. Find the area of the triangle.

3. Now find the area of the trapezium.

##### Question 6

Find the area of the trapezium shown.

## Kites

Kites are made up of two isosceles triangles. It is similar to a rhombus, but is not necessarily made up of two identical isosceles triangles, like a rhombus. This means a rhombus is a special type of kite.

The following applets explores how a kite has an area of half a rectangle.

Once again, by turning the kite into a rectangle, we can see that we end up with a rectangle that has dimensions equal to the diagonals of the kite.

Area of a kite
$\text{Area of a kite}=\frac{1}{2}\times\text{diagonal }x\times\text{diagonal }y$Area of a kite=12×diagonal x×diagonal y

#### Practice question

##### Question 7

The kite on the left can be split into two triangles as shown.

1. Find the area of one of the triangles.

2. Now find the area of the kite.

## Summary of area formulae

Shape Area Formula Dimensions
Rectangle

$\text{Area of a Rectangle}=\text{length }\times\text{width }$Area of a Rectangle=length ×width

or

$A=lb$A=lb

Square

$\text{Area of a Square}=\text{side }^2$Area of a Square=side 2

or

$A=s^2$A=s2

Parallelogram

$\text{Area of a Parallelogram}=\text{base }\times\text{height }$Area of a Parallelogram=base ×height

or

$A=bh$A=bh

Rhombus

$\text{Area of a rhombus }=\frac{1}{2}\times\text{diagonal }x\times\text{diagonal }y$Area of a rhombus =12×diagonal x×diagonal y

or

$A=\frac{1}{2}xy$A=12xy

Kite

$\text{Area of a kite}=\frac{1}{2}\times\text{diagonal }x\times\text{diagonal }y$Area of a kite=12×diagonal x×diagonal y

or

$A=\frac{1}{2}xy$A=12xy

Trapezium

$\text{Area of a Trapezium}=\frac{1}{2}\times\left(\text{side a}+\text{side b}\right)\times\text{height h}$Area of a Trapezium=12×(side a+side b)×height h

or

$A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h

## Finding an unknown dimension

Now that we are familiar with all the formulas, we can find a missing dimension if we know the area of the shape and the other dimensions needed to substitute into the area formula.

For instance, if we know the area of a rhombus and the length of one diagonal, we can substitute the known values in to the area formula and then rearrange to solve for the unknown dimension.

#### Practice question

##### Question 8

The kite below has an area of $48$48 cm2. The length of one of its diagonals is $12$12 cm.

1. If the other diagonal has a length of $k$k cm, solve for the value of $k$k.

Enter each line of working as an equation.