Virginia SOL 8 - 2020 Edition
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5.07 Review: Perimeter and circumference
Lesson

Perimeter

The perimeter of a shape is the length of its outline. For example, the perimeter of a rectangle is the sum of the lengths of all its four sides and the the perimeter of a triangle is the sum of the lengths of all its three sides.

Perimeters are useful for when we know the dimensions of a shape but not the distance around it, or maybe we know the distance around it and want to find its dimensions.

 

Perimeter of a rectangle

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

Perimeter of a rectangle

The perimeter of a rectangle has the formula: Perimeter $=$=$2$2$\times$×$($(Length $+$+ Width$)$)

$P=2l+2w$P=2l+2w

 

Perimeter of a square

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

Perimeter of a square

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

$P=4l$P=4l

Let's see these formulae in action.

Worked example

question 1

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.

 

Question 2

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.

 

Perimeter of regular polygons

Regular polygons are special in that all of their sides are the same length. A regular pentagon has five equal sides, a regular octagon has eight equal sides, and so on. This property of regular polygons makes it quite simple to find their perimeter.

Consider a regular hexagon.

Since it has six sides of equal length we can write its perimeter as:

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

Or more simply:

Perimeter $=$=$6$6$\times$×Length

In fact, we can do the same thing for any regular polygon.

Perimeter of a regular polygon

The perimeter of a regular $n$n-gon (a polygon with $n$n-sides) has the formula:

Perimeter $=$=$n$n$\times$×Length

 

Practice questions

Question 3

Find the perimeter of the shape given.

Question 4

This shape has a perimeter of $72$72 cm. What is the length of each side?

Question 5

Find the missing width of the rectangle $n$n.

 

Circumference

The circumference of a circle is the perimeter of a circle. As we have seen previously, it can be calculated with a special formula. 

Circumference of a circle

 

$C=\pi d$C=πd

and because the diameter is twice the radius, we can also write the formula as

$C=2\pi r$C=2πr

 

Estimations of pi

When we first learned about pi, we used both estimates and a calculator to help us. $\pi$π is an irrational number which is non-terminating and non-repeating, so sometimes we will use approximations for $\pi$π.

When calculating the circumference or area of circle, we will use pi. There are three options, but be sure to read the question to see if one option is required.

Approximations for pi
  • We can use the $\pi$π button on our calculator to get the most accurate value for $\pi$π
  • We will sometimes use $3.14$3.14 as an approximation for $\pi$π 
  • We will sometimes use $\frac{22}{7}$227 as an approximation for $\pi$π

 

Practice Questions

QUESTION 6

Find the circumference of the circle shown, correct to $2$2 decimal places. Use the value of $\pi$π from your calculator in your calculations.

QUESTION 7

If the radius of a circle is $98$98 cm, find its circumference to the nearest integer using the approximation $\pi=\frac{22}{7}$π=227.

QUESTION 8

What is the circumference of the Ferris wheel?

Give your answer correct to one decimal place. Use the value of $\pi$π from your calculator in your calculations.

 

Problem solving with circular shapes

If you take a look around, you should be able to spot some objects that have circular features. A clock face, a round plate, or the lid of a jar are some examples of things that have the outline of a circle.

For lots of other common objects there are circles present in different parts of their shape. Some examples include the rim of a cup, or the crust on a pizza.

Once we have found a feature of an object that looks like a circle, we can explore the relationship between the radius, the diameter, and the circumference of that circle. In his way we can understand more about the overall object.

 

Remember!

The diameter $d$d of a circle is equal to twice the radius $r$r, so $d=2r$d=2r.

If we know the diameter of a circle, the circumference $C$C is given by the equation $C=\pi d$C=πd.

If we know the radius of a circle, the circumference is given by the equation $C=2\pi r$C=2πr.

 

Worked example

Question 9

Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of $5$5 m. What distance does Lisa cover if she walks all the way around the pool? Use the $\pi$π button on your calculator and give your answer to one decimal place.

Think: The distance around the outside of a circle is its circumference. We can use the radius of the pool to find its circumference.

Do: Using the equation $C=2\pi r$C=2πr, we substitute $r=5$r=5 m.

$C$C $=$= $2\pi r$2πr

Formula for circumference

  $=$= $2\pi\times5$2π×5

Substituting in $r=5$r=5

  $=$= $10\pi$10π

Simplifying partially

  $=$= $31.41592$31.41592$...$...

Evaluating on a calculator

  $=$= $31.4$31.4

Rounding correctly

 

Lisa will walk $31.4$31.4 m around the pool.

Reflect: Suppose Lisa's neighbor has a circular pool with twice the circumference. What would be its radius?

 

Circular motion

We can also use these same ideas about circles to understand the circular motion of objects. Think about the way the end of the second hand moves around a clock, or the way a wheel rolls down a hill.

Worked example

Question 10

Harry pushes a hula hoop down a hill, and it rolls all the way to the bottom. The hula hoop has a diameter of $85$85 cm. If the hula hoop rolled a total of $25$25 m, how many revolutions did it make on its way down the hill? Use $\frac{22}{7}$227 as an approximation for $\pi$π and give your answer to two decimal places.

Think: The hula hoop will make one full revolution when it rolls a distance equal to its circumference. We can use the diameter to find the circumference, then divide the total distance the hoop rolled by this length to find the number of revolutions.

We have both cm and m, so let's convert them to the same unit.

Do: If we work in m, the diameter of the hula hoop is $0.85$0.85 m.

$C$C $=$= $\pi d$πd

Formula for circumference

  $=$= $\frac{22}{7}\times0.85$227×0.85

Substitute in the diameter and our approx. $\pi$π

  $=$= $\frac{22}{7}\times\frac{85}{100}$227×85100

Convert $0.85$0.85 to a fraction

  $=$= $\frac{187}{70}$18770

Multiply and simplify the product


Now we can divide the total distance rolled by this circumference.

Number of revolutions $=$= $\frac{\text{total distance rolled }}{\text{circumference }}$total distance rolled circumference
  $=$= $25\div\frac{187}{70}$25÷​18770
  $=$= $25\times\frac{70}{187}$25×70187
  $=$= $\frac{1750}{187}$1750187
  $=$= $9.36$9.36

 

The hula hoop has rolled about $9.36$9.36 times as it covers the distance of $25$25 m.

Reflect: How different would the answer be if we used the $\pi$π button on our calculator? How many decimal places would we need to round to in order to see a difference?

 

Practice questions

Question 11

A bicycle tire has a diameter of $34$34 cm. What is the circumference of the tire?

Give your answer correct to one decimal place. Use the value of $\pi$π from your calculator.

Question 12

What is the length of the strip of  seaweed around the outside of the sushi?

Give your answer correct to one decimal place. Use the value of $\pi$π from your calculator.

 

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