## Thursday, November 22, 2012

### AREA

## Area of a SquareThe area of a square that has a side-length of 1 cm is 1 square centimetre (or 1 cm ^{2}). To find the area of a square by the method of counting squares, we divide the square into smaller squares of one centimetre side-length .Consider a square that has a side-length of 4 cm. Using the method of counting squares, we find that the area of the square is 16 cm ^{2}.Clearly, the square contains 4 rows of 4 squares. Therefore: ##
The area of a square is equal to its side-length multiplied by its side-length. That is: | |

##
To find the area of a region enclosed within a plane figure, draw a diagram and write an appropriate formula. Then substitute the given values and use a calculator, if necessary, to obtain the required area. |

Perimeter |

A square and an equilateral triangle are both examples of regular polygons. Another method for finding the perimeter of a regular polygon is to multiply the number of sides by the length of one side. Let's revisit Examples 3 and 4 using this second method. | ||

Example 3: | Find the perimeter of a square with each side measuring 2 inches. | |

Solution: | This regular polygon has 4 sides, each with a length of 2 inches. Thus we get: | |

= 4(2 in) = 8 in | ||

Example 4: | Find the perimeter of an equilateral triangle with each side measuring 4 centimeters. | |

Solution: | This regular polygon has 3 sides, each with a length of 4 centimeters. Thus we get: | |

= 3(4 cm) = 12 cm | ||

Example 5: | Find the perimeter of a regular pentagon with each side measuring 3 inches. | |

Solution: | = 5(3 in) = 15 in | |

Example 6: | The perimeter of a regular hexagon is 18 centimeters. How long is one side? | |

Solution: | = 18 cm | |

Let represent the length of one side. A regular hexagon has 6 sides, so we can divide the perimeter by 6 to get the length of one side (). | ||

= 18 cm ÷ 6 | ||

= 3 cm | ||

Summary: | To find the perimeter of a polygon, take the sum of the length of each side. The formula for perimeter of a rectangle is: . To find the perimeter of a regular polygon, multiply the number of sides by the length of one side. |

Perimeter Part I |

The perimeter of a polygon is the distance around the outside of the polygon. A polygon is 2-dimensional; however, perimeter is 1-dimensional and is measured in linear units. To help us make this distinction, look at our picture of a rectangular backyard. The yard is 2-dimensional: it has a length and a width. The amount of fence needed to enclose the backyard (perimeter) is 1-dimensional. The perimeter of this yard is the distance around the outside of the yard, indicated by the red arrow; It is measured in linear units such as feet or meters. | ||

To find the perimeter of a polygon, take the sum of the length of each side. The polygons below are much smaller than a fenced-in yard. Thus, we use smaller units in our examples, such as centimeters and inches. | ||

Example 1: | Find the perimeter of a triangle with sides measuring 5 centimeters, 9 centimeters and 11 centimeters. | |

Solution: | P = 5 cm + 9 cm + 11 cm = 25 cm | |

Example 2: | A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the perimeter. | |

Solution 1: | P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm | |

Solution 2: | P = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm | |

In Example 2, the second solution is more commonly used. In fact, in mathematics, we commonly use the following formula for perimeter of a rectangle: , where is the perimeter, is the length and is the width. | ||

In the next few examples, we will find the perimeter of other polygons. | ||

Example 3: | Find the perimeter of a square with each side measuring 2 inches. | |

Solution: | = 2 in + 2 in + 2 in + 2 in = 8 in | |

Example 4: | Find the perimeter of an equilateral triangle with each side measuring 4 centimeters. | |

Solution: | = 4 cm + 4 cm + 4 cm = 12 cm |

**Perimeter worksheet 1**

**Remember**:

**Perimeter is the total distance around the outside of a 2D shape.**

**You calculate it by adding together all the lengths of a shape.**

Work out the perimeter of these shapes:

1. A rectangle 7cm long by 8cm wide.

2. A square with one side 15cm long.

3. A rectangle 20cm by 30cm.

4. A 32cm by 10cm rectangle.

5. A rectangle 75cm long by 25cm wide.

6. A 50cm by 50 cm square.

7. A rectangle 16cm by 12cm.

8. A 63cm by 25cm rectangle.

9. A square with one side measuring 45cm.

10. A rectangle 19cm long by 5cm wide

Exercise 2

Find the area and perimeter of each rectangle.

Remember to include the units in your answers.

(For example, m or square m)

1. 18 m wide, 23 m long

perimeter =

area =

2. 39 in wide, 30 in long

perimeter =

area =

3. 10 ft wide, 15 ft long

perimeter =

area =

4. 23 cm wide, 21 cm long

perimeter =

area =

5. 27 mm wide, 10 mm long

perimeter =

area =

6. 13 m wide, 46 m long

perimeter =

area =

7. 11 ft wide, 31 ft long

perimeter =

area =

8. 49 in wide, 8 in long

perimeter =

area =

9. 16 cm wide, 29 cm long

perimeter =

area =

10. 40 mm wide, 16 mm long

perimeter =

area =

### HOW TO FIND THE ARES

To get the amount of space inside a figure, we use a square to represent 1 unit and we say that the area is measured in square units

Take a look at the following rectangle. To get the area, we are going to draw squares of equal sizes inside of it.

1 square represents 1 square unit. The rectangle has 8 squares, so the area for this rectangle is 8 square units.

We can also write 8 units

Notice,it is very important, that you can get the same answer if you multiply 2 square units by 4 square units because 2 × 4 = 8

2 square units represent the measure of the width and 4 square units represent the measure for the length.

Thus, in general, to get the area for a rectangle, just use the following formula:

Take a look at the following rectangle. To get the area, we are going to draw squares of equal sizes inside of it.

1 square represents 1 square unit. The rectangle has 8 squares, so the area for this rectangle is 8 square units.

We can also write 8 units

^{2}and it will mean the sameNotice,it is very important, that you can get the same answer if you multiply 2 square units by 4 square units because 2 × 4 = 8

2 square units represent the measure of the width and 4 square units represent the measure for the length.

Thus, in general, to get the area for a rectangle, just use the following formula:

**Area of rectangle = length × width**
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