6. Geometry & Measurement

Lesson

The perimeter of a shape is the measure of the distance around a figure. Think about walking around the outside of a school, that distance is the school’s perimeter. A way to remember is to think “peRIMeter” because the edge of an object is its rim and peRIMeter has the word “rim” in it.

Suppose a farmer wants to make a rectangular paddock for his sheep. He has decided that the paddock must be $20$20 m long and $15$15 m wide, and needs to determine how much fencing to purchase.

The farmer can find out the required length of fencing by finding the perimeter of the rectangle enclosed by the fence.

We can see that the rectangular paddock will have two sides of length $20$20 m and two sides of length $15$15 m.

Adding these four sides together will give us the perimeter.

Perimeter | $=$= | $20+20+15+15$20+20+15+15 |

$=$= | $70$70 |

So the farmer will need $70$70 m of fencing to make his paddock.

Every rectangle has two pairs of equal sides, which we can call the length and the width.

As we can see from the image, the perimeter of a rectangle will always be:

Perimeter $=$= Length $+$+ Width $+$+ Length $+$+ Width

This is the same as two groups of $($(Length $+$+ Width$)$), which we can write as:

Perimeter $=$=$2$2$\times$×$($(Length $+$+ Width$)$)

Perimeter of a rectangle

The perimeter of a rectangle has the formula:

Perimeter $=$=$2$2$\times$×$($(Length $+$+ Width$)$)

or

Perimeter $=$= $2\times$2×Length $+$+ $2\times$2×Width

The main property of a square that we can use to calculate its perimeter is that it has four equal sides.

As we can see from the image, the perimeter of a square will always be:

Perimeter $=$= Length $+$+ Length $+$+ Length $+$+ Length

This is the same as four groups of the Length, which we can write as:

Perimeter $=$=$4$4$\times$×Length

Perimeter of a square

The perimeter of a square has the formula: Perimeter $=$=$4$4$\times$×Length

Let's see these formulas in action.

A rectangular race track has a length of $42$42 m and a width of $8$8 m. How long is one lap?

**Think:** One lap of the race track is equal to its perimeter. Since the race track is a rectangle, we can use the formula,

Perimeter $=$=$2$2$\times$×$($(Length $+$+ Width$)$).

**Do:** We can find the perimeter of the rectangle by substituting the dimensions of the track into the formula. This gives us:

Perimeter | $=$= | $2\times\left(42+8\right)$2×(42+8) | (Substitute the length and width of the track) |

$=$= | $2\times50$2×50 | (Sum the numbers in the parentheses) | |

$=$= | $100$100 | (Perform the multiplication) |

So we find that one lap of the race track is $100$100 m long.

**Reflect:** First, we identified that one lap was equal to the perimeter of the race track, then we applied the formula for the perimeter of a rectangle to find the lap length.

A $32$32 cm long piece of wire is bent into the shape of a square. What is the side length of the square?

**Think:** Since the piece of wire was $32$32 cm long, we know that the perimeter of the square will be $32$32 cm. We can try reversing the formula for the perimeter of a square, Perimeter $=$=$4$4$\times$×Length, to find a solution.

**Do:** We know that $32$32 cm $=4\times$=4×Length, so we can find the number that multiplies with $4$4 to give $32$32. This can be found by dividing the perimeter by the number of sides, which gives $\frac{32}{4}=8$324=8.

So we find that the side length of the square is $8$8 cm.

**Reflect:** First, we identified that the square had a perimeter equal to the length of the wire, then we reversed the formula for the perimeter of a square to find the side length.

Find the perimeter of the rectangle shown.

Find the perimeter of the square shown.

Nadia wants to build a thin wire frame for a photo that is $13$13 cm long and $6$6 cm wide. How much wire will she need to go around the entire photo?

The area of a rectangle is the amount of space that can fit within its outline. One method of finding the area is to divide a rectangle into unit squares and to count the number of these squares.

A unit square is defined to be a square with a side length of $1$1 unit, and a single unit square has an area of $1$1 unit^{2}. In this way, counting the number of unit squares in a shape tells us the area of that shape as a multiple of $1$1 unit^{2}.

The rectangle below has length $4$4 cm and width $2$2 cm and is divided into unit squares, each with a side length of $1$1 cm. Since the units of the dimensions are in cm, the area will be expressed in cm^{2}.

By counting, we can see there is a total of $8$8 unit squares. So this rectangle has an area of $8$8 cm Can you see how we can use the dimensions $4$4 cm and $2$2 cm to find the area? |
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Area of rectangle

The area of a rectangle has the formula: Area $=$= Length $\times$× Width or $A=l\times w$`A`=`l`×`w`

Similarly, this method can be used to find the area of a square. Below, a square with side length $5$5 m is divided into unit squares, each with a side length of $1$1 m. Since the units of the dimensions are in m, the area will be expressed in m^{2}.

By counting, we can see there is a total of $25$25 unit squares. So this square has an area of $25$25 m Can you see how we can use the dimensions of the square to find its area? |
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Area of a square

The area of a square has the formula: Area $=$= side $\times$× side or $A=s^2$`A`=`s`2

Can rectangles look differently but have the same area? Noticed that multiplying the dimensions of a rectangle gives its area and multiplication is a communicative property. Therefore, the length and the width can be swamped and produce the same result.

Is there more than two numbers that multiply to equal $36$36? Yes,

$1\times36=36$1×36=36

$2\times18=36$2×18=36

$3\times12=36$3×12=36

$4\times8=36$4×8=36

$6\times6=36$6×6=36

Therefore, all of these combinations can produce rectangles with the same area. In other words, it possible for rectangles to have the same area when they have different dimensions.

Use the applet below to draw rectangles with a target area. The left side of the applet will tell you how many rectangles can be drawn. Once you have drawn the required number of rectangles, click $\editable{\text{Next game}}$Next game.

As an added challenge, you can click $\editable{\text{Hide Grid}}$Hide Grid to draw rectangles by thinking of two numbers that multiply to give the target area, rather than counting the unit squares.

Find the area of the rectangle below.

**Think**: We know the length and width of the rectangle, so we can use the formula to find the area. The dimensions are given in mm, so the area will be in mm^{2}.

**Do**:

$\text{Area }$Area | $=$= | $\text{length }\times\text{width }$length ×width | (Formula for the area of a rectangle) |

$=$= | $5\times3$5×3 | (Substitute the values for the length and width) | |

$=$= | $15$15 | (Perform the multiplication to find the area) |

So this rectangle has an area of $15$15 mm^{2}.

**Reflect**: The length of a rectangle is often defined to be the longest side, which makes the width the shortest side. But since the area formula uses multiplication, the order does not matter. So even if we use another convention that defines the length to be always the horizontal side, the same formula will give the same area.

Find the area of the square below.

**Think**: We know the side length of the square, so we can use the formula to find the area. The dimension is given in cm, so the area will be in cm^{2}.

**Do**:

$\text{Area }$Area | $=$= | $\text{side }\times\text{side }$side ×side | (Formula for the area of a square) |

$=$= | $6\times6$6×6 | (Substitute the value for the side) | |

$=$= | $36$36 | (Perform the multiplication to find the area) |

So this square has an area of $36$36 cm^{2}.

**Reflect**: The area of a square involves the product of the side length with itself. Another way to write this is using exponential notation, $\text{Area }=\text{side }^2$Area =side 2. In this example, the area would be $6^2=36$62=36 cm^{2}.

Find the area of the rectangle shown.

Find the area of the square shown.

A basketball court is $29$29 m long and $15$15 m wide.

What is the area of the basketball court?

Solve problems, including practical problems, involving area and perimeter of triangles and rectangles