Introduction

Around the world, street performers delight audiences with their up-close and engaging shows. As shown here in Covent Gardens in London, performers often lay out rope to keep audience members outside of their performing space.

Imagine you are a street performer and you need to lay out a red rope to mark-off your stage. Pick up the red rope below and create a stage on the cobblestones so that:

Now, let’s focus on the shape we made out of the rope and imagine it as a shape drawn on a piece of paper. Recall that perimeter is the length of the boundary, or outside, of a shape, and area is the how much space a shape covers or encloses.

You used the same different amount of rope each time, but created shapes with different the same number of cobblestones roped off. Different shapes with the same different perimeter can have different the same areas.

Now, let’s work on the difference between perimeter and area of a shape. Let’s think through five examples that highlight this difference.

Football clubs often need to replace the grass field after it has been worn down. In 2018, the Estadio Azteca in Mexico City looked like this before the field was replaced.

Clubs often choose between squares of grass or artificial turf to replace the worn down grass:

Below is a picture of a worn out football field. Drag new squares of grass onto the worn out field below to model replacing the grass.

Let’s make an estimate of how many squares of grass we’ll need to replace the field. Just look at the picture of the field and make a quick guess. Enter the guess here: [TODO]

Recall your estimate of the area perimeter of the field is TODO square meters meters square centimeters. We’ll find out later in this chapter how close your estimate is!

Now that the new grass is in place, we need to paint the lines on the field.

Drag each strip of white onto the field to model painting a while line around the field. One white strip equals one can of paint.

Make an estimate of how many cans of paint we’ll need to paint a line around the outside of the field. Just look at the picture of the field and make a quick guess. Enter the guess here: [TODO]

Each can of paint can make a line 1 meter long. Recall your estimate of the perimeter area of the field is TODO meters square meters centimeters. Again, we’ll find out later how close your estimate is.

Now, let’s move on to thinking more deeply about the area of rectangles. Below are a bunch of square centimeters. Remember, these are squares whose sides lengths are each centimeter long. Drag square centimeters into the rectangle so you fill in the rectangle.

This rectangle has rows, and every row has squares, so we used 3x5= squares in total.

Each square is one square centimeter, so the area perimeter of this rectangle is square centimeters centimeters square meters.

Again, the length of the side of each square is centimeter. So we know, the length of the base of the rectangle is centimeters and the length of the height of the rectangle is centimeters. The perimeter of the rectangle is centimeters square centimeters.

Before we continue on with area and perimeter, let’s discuss some commonly used units for perimeter and area. In this chapter, we’ll use centimeters, meters and kilometers.

A centimeter is about the width of a staple length of a guitar distance of about two laps around a track.

A meter is about the length of a guitar width of a staple distance of about two laps around a track.

A kilometer is about distance of about two laps around a track length of a guitar width of a staple.

Area is the amount of space inside an object., To determine the area of a shape, we’ve been filling up the space inside an object with squares. We could use other shapes as well. You can study that idea in a chapter on tessellations.

Square meters are squares with side lengths of meter centimeter kilometer, and square centimeters are squares circles triangles with side lengths of centimeter meter kilometer. So, a square kilometer is a square with side lengths of kilometer meter centimeter.

When we talk about a certain number of square meters, say 7 of them, we can write it as “7 square meters” or “7 m2”. In fact, the “m2" is read as “meters squared.” Any power of “2” is often read as “squared.” For example, we can read 92 as “9 squared” because we would need to multiply 9x9 when finding the area perimeter of a square with side length 9.

Sort the units of measure below into those that measure length and those that measure area:

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Now, sort the units of measure for area. Each unit of measure is represented in three different ways. Drag each of the labels into the correct square.

Below is a square meter and a bin of squares that are 12 meter on each side. See how many 12 square meters you need to fill in the square meter:

It takes 412414434 square centimeters to fill in 1 square meter. So, the area of a square with a 12 meter on each side is of the area of 1 square meter.

Let’s apply our understanding of area to another example. Begin exploring the area of the rectangle below by dragging in any 3 of the area shapes on the left into the blue rectangle.

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Let’s see if we can determine the area of the blue rectangle. The base is centimeters long and the height is centimeters long. The area of the rectangle is 834812814 square centimeters centimeters square meters.

The area of the rectangle is 834 square centimeters.

Our first area example has 3 rows of 5 square centimeters each for a total of square centimeters centimeters. 3x5=.

This example has 212 214 234 rows of 312 square centimeters. So, to find the area we could do 2121434x3121434 which does indeed equal 834812814.

If you forget how to calculate 212x312, you can review the chapter on Multiplying and Dividing Fractions. 212x312 means 212 groups of 312.

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This arrangement looks familiar! 212x312 = 834812814.

So, by counting the number of rows and number of columns, and multiplying adding subtracting dividing those numbers, we have found the formula for the area of any rectangle circle triangle.

Area of Rectangle = number of units along the base x + - number of units along the height diagonal. We can abbreviate this formula as:

Area of Rectangle = base x + - height base diagonal.

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Let’s think about the perimeter of this same rectangle. Below are some calculations using the side lengths of the rectangle above. Sort the calculations into those that will give you the correct perimeter and those that will not.

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The perimeter of the rectangle is centimeters square meters square centimeters.

Let’s end this chapter by coming back to our work with the football field. You guessed the area of the field to be TODO square meters meters. Below is the image of the field as well as a picture of your estimation:

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It seems that your estimate of the area was [[too big | too small].

Now, let’s find the actual area of the football field.

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The length of the base is meters square meters and the length of the height is meters square meters. This means, we would need rows of pieces of square meter grass to cover the field. To calculate the area, we need to multiply 75x110. This gives us an area of square meters meters. Your estimate was TODO square meters away.

Let’s revisit your perimeter guess. You guessed the perimeter of the field to be TODO meters square meters. Below is a picture of the field as well as a field with your estimated perimeter.

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