 # 9.01 Areas of special quadrilaterals

Lesson

## Introduction

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

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

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.

### Exploration

The following applet explores these similarities further.

A parallelogram can be rearranged to form a rectangle with the same base and perpendicular height.

The area of a parallelogram can be found by multiplying the base and the perpendicular height, same as a rectangle.

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

### Examples

#### Example 1

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

Worked Solution
Create a strategy

Use the formula to find the area of the rectangle.

Apply the idea

Any parallelogram can be rearranged to form a rectangle:

The width of the rectangle is equal to the perpendicular height of the parallelogram, and their base lengths are the same.

Idea summary

The area of a parallelogram can be found using the formula:

\displaystyle A=bh
\bm{b}
is the base of the parallelogram
\bm{h}
is the perpendicular height

## 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.

### Exploration

The following applet explores these similarities further

A rhombus can be rearranged to form a rectangle.

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

### Examples

#### Example 2

Find the area of the rhombus shown.

Worked Solution
Create a strategy

We can find the area of rhombus using the formula: A=\dfrac{1}{2}xy

Apply the idea

We can see from the diagram that the diagonals have lengths of 4 mm and 8 mm.

Idea summary

The area of a rhombus can be found using the formula:

\displaystyle A=\dfrac{1}{2}xy
\bm{x}
is the length of one diagonal
\bm{y}
is the length of the other diagonal

## 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.

### Exploration

Use following applet to explore how the area of a trapezium and a parallelogram relate to each other.

The area of a trapezium is exactly half of the area of a parallelogram that has the same height and a base equal to the total length of the trapeziums parallel sides.

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, will be the average length of the two parallel sides, 6 and 13, so 9.5 will be the length of the rectangle formed.

Area of a trapezium is given by: \text{Area} = \dfrac{1}{2}\times \left(\text{side }a+\text{side }b\right)\times \text{height }h or A=\dfrac{1}{2}(a+b)h.

The sides a and b are the lengths of the parallel sides, and h is the perpendicular height.

### Examples

#### Example 3

Find the area of the trapezium shown.

Worked Solution
Create a strategy

We can find the area of trapezium using the formula:A= \dfrac{1}{2}(a+b)h

Apply the idea

The side lengths are a=5\ \text{cm} and b=8\ \text{cm}, and the height is h=3 cm.

Idea summary

The area of a trapezium is given by:

\displaystyle A=\dfrac{1}{2}(a+b)h
\bm{a}
is a parallel side length
\bm{b}
is the other parallel side length
\bm{h}
is the perpendicular height

## 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.

### Exploration

The following applet 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 is given by: \text{Area of a kite}=\dfrac{1}{2}\times \text{diagonal }x\times \text{diagonal }y or A=\dfrac{1}{2}xy.

### Examples

#### Example 4

Find the area of the kite below.

Worked Solution
Create a strategy

We can use the formula: A=\dfrac{1}{2}xy.

Apply the idea
Idea summary

A summary of the area formulas for quadrilaterals:

## Unknown dimensions

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.

### Examples

#### Example 5

The kite below has an area of 48 \text{ cm}^2. The length of one of its diagonals is 12\,cm.

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

Worked Solution
Create a strategy

Substitute the area, the known diagonal, and the pronumeral k into the formula for the area of a kite.

Apply the idea
Idea summary

We can find a missing dimension of a quadrilateral if we know the area and the other dimensions. We just substitute these values into the area formula and solve for the unknown dimension.

### Outcomes

#### MA4-12MG

calculates the perimeters of plane shapes and the circumferences of circles

#### MA4-13MG

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

#### MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles