# Area Of Quadrilaterals

**Area** is calculated when we want to find the **flat surface** to:

- collect sunshine for
**solar panels** **paint**a wall**tile**a bathroom**carpet**a floor**spread grass seeds**to grow a lawn**fertilize**a garden

A **Quadrilateral** is a four-sided shape such as a rectangle, a square, a parallelogram and a trapezoid.

The area rules for each of these is based on the Area of Rectangle rule and are:

Area of Rectangle = L × W

Area of Square = S × S

Area of Parallelogram = L × H

Area of Trapezoid = | (A + B) | × H |

2 |

**CONVERTING AREA UNITS**

To change:

- square centimetres (cm
^{2}) to square millimetres (mm^{2}), multiply by 10 and then multiply again by 10 (or multiply by 100). - square metres (m
^{2}) to square millimetres (mm^{2}), multiply by 1000 and by 1000 again (or multiply by 1 000 000). - square metres (m
^{2}) to square centimetres (cm^{2}), multiply by 100 and by 100 again (or multiply by 10 000). - square kilometres (km
^{2}) to square metres (m^{2}), multiply by 1000 and by 1000 again (or multiply by 1 000 000).

**IF YOU SQUARE THE NAME, YOU SQUARE THE NUMBER.**

## Example One - Area of Rectangle

The International Space Station has been continuously occupied by astronauts performing experiments in astronomy, biology, chemistry and physics for over 12 years.
It has four double-sided photo-voltaic arrays (solar panels) to make electricity. **Each of the 8 panels measures 58 metres long by 3 metres wide.**
What is its total sunlight collection area?

**Answer:**

Area of Rectangle

= L × W

= 58 × 3

= 174 square metres or 174 m^{2}

Area of 8 Rectangular Panels

= 174 × 8

= 1392 square metres or 1392 m^{2}

## Did You Know That...?

The commonly used world map is misleading when displaying the **actual area of countries**. The **Gall-Peters Map Projection** shown above is more accurate.

## Example Two - Area of Square

What is the tiled area of a square bathroom floor with a **side of 3000mm**?
*(Remember carpenters measure accurately in millimetres, but tiles are priced by the square metre.)*

**Answer:**

Area of Square

= S × S

= 3 × 3

= 9 m^{2}

## Example Three - Area of Parallelogram

One thousand postage stamps in the shape of a parallelogram are printed. Each stamp has a length of **25 mm** and a
perpendicular height (at right angles to the length) of **12 mm**. What is the area of glue (on the reverse side) to cover **1000 stamps**?

**Answer:**

Area of Parallelogram

= L × H

= 25 × 12

= 300 mm^{2}

Area of glue on 1000 stamps

= 300 × 1000

= 300 000 mm^{2} or 3000 cm^{2}

## Example Four - Area of Trapezoid

A farm in the shape of a trapezoid and has **parallel sides of 1000 metres and 2000 metres** respectively. The **perpendicular height is 500 metres**.
What is the area (in hectares) of this property? *(Hint: 1 hectare = 10 000 square metres)*

**Answer:**

Area of Trapezoid = | (A + B) | × H |

2 |

= | (1000 + 2000) | × 500 |

2 |

= 1500 × 500

= 750 000 m^{2}

= 75 hectares

## Example Five - Painting a Wall

Calculate the quantity of paint required to paint a **rectangular bedroom wall 4 metres by 2.4 metres**.
**Two coats of paint** are needed. **One litre of paint covers 10m ^{2}**.

**Answer:**

Area (1 coat)

= L × W

= 4 × 2.4

= 9.6 m^{2}

Area (2 coats)

= 9.6 × 2

= 19.2 m^{2}

Number of litres

= 19.2 / 10

= 1.92

= 2 litres *(Remember that you can't buy parts of litres of paint from the hardware store.)*

## Questions

**Find the areas of the following quadrilaterals. Be careful to convert the dimensions to the same units first.**

**Q1.** Square with a side of 7 metres
**Q2.** Rectangle with a length of 10km and a width of 700 metres
**Q3.** Parallelogram with a length of 5 metres and a perpendicular height of 60cm
**Q4.** Trapezoid with parallel sides of 10cm and 6cm, and a perpendicular height of 5cm

**Answers**
**A1.** 49 m^{2}
**A2.** 21.4 km^{2} or 7000000 m^{2}
**A3.** 3 m^{2} or 30000 cm^{2}
**A4.** 5 cm^{2}

## Maths Fun - Trapezoidal Numbers

**Trapezoidal Numbers** (also called **Polite Numbers**) are the sum of consecutive positive integers. Examples of Trapezoidal Numbers are:

- 1 + 2 = 3
- 1 + 2 + 3 = 6
- 2 + 3 + 4 = 9
- 3 + 4 + 5 + 6 = 18
- 10 + 11 + 12 + 13 = 46

Why is the number 4 "impolite"? Why do you think these are called "Trapezoidal"? (Hint: Think of arranging blocks like steps.)