10.5 Perimeters

Imagine you were decorating a room and you decided to put a border along the top of the walls. To determine the length of the border, you would need to measure the lengths of the walls and add these measurements. For example, if the room is rectangular and measures 5 m by 4 m, the length of the border would be five m plus four m plus five m plus four m or 18 m, as shown in the following diagram. [ Make sure all the lengths are measured in the same unit! ]

The distance around the edge of a shape is known as the perimeter. The prefix “peri” means around and “meter” means measure, so “perimeter” means measuring around a shape. For a shape with straight edges, you can work out the perimeter by measuring the length of each edge and then adding these lengths.

Activity Symbol Activity: Around the Edge

By measuring lengths on your tangram puzzle, work out the perimeter of the following: [ Using all seven tangram pieces, make one shape that has the largest perimeter and one shape that has the smallest perimeter. Can you prove your results? ]

(a) The square ABCD

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Answer

(a) ABCD is a square whose sides are 10 cm. So the perimeter is 10 cm plus 10 cm plus 10 cm plus 10 cm equals 40 cm.

(b) prefix white up pointing triangle of cap f times cap g times cap i

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Answer

(b) In the prefix white up pointing triangle of cap f times cap g times cap i, cap g times cap i equals five cm, cap g times cap f almost equals 3.5 cm, and cap f times cap i almost equals 3.5 cm. Adding these lengths gives the perimeter as approximately 12 cm.

(c) The trapezoid DGIH

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Answer

(c) DH and GI are each 5 cm long; DG is approximately 10.6 cm and HI is approximately 3.5 cm. So the perimeter of DGIH is 10.6 cm plus five cm plus three .5 cm plus five cm equals 24 .1 cm.

(d) This figure formed from shapes 3, 4, and 5:

This shape has been formed by putting the square (piece 5) along the shorter side of the parallelogram (piece 3). One of the shorter sides of the triangle (piece 4) has been put next to the side of the square so that the long side of the parallelogram and the long side of the triangle lie in a straight line.
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Answer

(d) Starting on the left at the top of shape 3 and going clockwise around the figure, the perimeter is: five cm plus three .5 cm plus three .5 cm plus three .5 cm plus five cm plus five cm plus three .5 cm equals 29 cm.

When a shape has straight sides, you can measure the perimeter fairly easily by considering each side in turn. However, measuring lines that are not straight can be more difficult. If you need the measurement for some practical purpose, then you can use a piece of string to wrap around an object and then measure the string.

Activity Symbol Activity: Circles

You will need a tape measure (or a piece of string), a ruler, and five objects such as mugs, cans, bowls, or buckets that are cylindrical with circular tops. Measure the circumference and diameter of each object in centimeters to the nearest 0.1 cm. Use the tape measure to find the circumference and the ruler to find the diameter.

If you do not have a tape measure, use a piece of string and then measure the string using the ruler. You could also use a piece of graph paper to measure the circumference. Write down the diameter and the circumference of each object in a table and calculate the ratio circumference divided by diameter. What do you notice?

Answer

Some sample results:

Item Diameter (in cm) Circumference (in cm) circumference divided by diameter (rounded to 1 decimal place)
Spice jar 4.4 14.1 3.2
Drinking glass 6.6 20.9 3.2
Tin container 8.6 27 3.1
Mug 10.3 32.3 3.1
Bowl 23.0 72.8 3.2

It is difficult to measure these quantities accurately. However, it is noticeable that in each case, the ratio of the circumference to the diameter seems to be about 3.1 or 3.2. In other words, the circumference is just over three times the length of the diameter.

If you could measure these objects more accurately, you would find that the circumference divided by the diameter always gives the same answer. This value is known as “pi” (pronounced “pie”) and it is denoted by the Greek letter pi. The value of pi is approximately 3.142, although for most calculations, you will be using the The button shown is the pi key. button on your calculator, which shows pi to 14 decimal places. [ Pi occurs in many branches of mathematics, not just problems with circles! ]

Pi is an irrational number. This means that it can’t be represented exactly as a fraction and has an infinite number of decimal places with no repeating pattern of numbers.

Videoclip iconIf you would like to explore pi further and have a bit of fun finding where you, your family and friends’ birthdays occur within it, visit this website [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .

The following formula enables you to work out the circumference of any circle: circumference equation left hand side equals right hand side pi postfix multiplication diameter.

Because we know that the diameter is twice the radius, this can also be written as:

circumference equation left hand side equals right hand side two multiplication pi postfix multiplication radius

Notepad iconActivity: Circumference of a Circle

Suppose a circular table has a diameter of 1.5 meters. How many people can sit down comfortably for a meal at the table, assuming that each person requires a space of about 0.75 m?

This is a circle with the diameter marked as 1.5 m.
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Answer

We know the diameter, so the circumference can be solved with the formula: circumference equation left hand side equals right hand side pi postfix multiplication diameter.

As we just require a rough estimate here, we can substitute 3.142 for pi and 1.5 for the diameter. This gives the following solution (rounded to 1 decimal place): circumference multirelation almost equals 3.142 multiplication 1.5 equals 4.7 m.

So, the circumference of the table is about 4.7 m.

Assuming each person needs a space of about 0.75 m in width, the number of people who can fit around the table is 4.7 division 0.75 almost equals six. Therefore, six people should be able to fit around the table.

(If you don’t require a rough estimate though you should use the  button on your calculator).

10.4.2 What Can You See?

10.5.1 Calculator Exploration: Introducing Pi