Lesson Video: Converting and Comparing Area: Metric Units Mathematics

Video Transcript

In this video, we’re going to look at how to convert between area units in the metric system. For example, how we change square centimetres or square decimetres into square metres or how we change square kilometres into square metres.

Before we begin with our area conversions, let’s recall some familiar metric length conversions. One centimetre is equal to 10 millimetres. One decimetre is equal to 10 centimetres. One metre is equal to 10 decimetres. And a metre is also equal to 100 centimetres. And one kilometre equals 1000 metres. We can order our units from the largest to smallest, the largest being kilometres and the smallest being millimetres.

If we had a unit in kilometres and we wanted to change it into metres, we could look at our conversion and say that we must multiply our number by 1000. From metres to decimetres, we would multiply by 10. From decimetres to centimetres, we would multiply by 10 and the same from going from centimetres to millimetres. If instead we wanted to go in the opposite direction, that is, changing from a small unit to a large unit, then we do the inverse operation. Here, we would be dividing by the power of 10, that is, dividing by 10 or 1000.

So let’s now see how we can use these length conversions when it comes to working out an area conversion. Let’s say we have a square of one metre by one metre. To work out the area, we’d multiply the length by the width. In this case, that would be one times one. So that’s one metre squared. But what if we took the same square and instead of measuring it in metres, we measured it as 100 centimetres. In this case, our area would be equal to 100 times 100 square centimetres, which is 10000 square centimetres. So since our squares have the same area, we now know that one square metre is equal to 10000 square centimetres.

To change between areas, we can use the quick technique of squaring the length conversion. We can remember, for example, that if we want to change metres into centimetres, we multiply by 100. If we want to go the opposite direction from centimetres to metres, we divide by 100. So if we square the length conversion, then if we want to change from square metres to square centimetres, we square the length conversion. So we would multiply by 100 squared, which is the same as multiplying by 10000. To go from square centimetres to square metres, we would divide by 100 squared, which is the same as dividing by 10000.

Let’s take a look at how we would convert square decimetres into square metres. Let’s take a look at our two equivalent squares. One of them is one metre by one metre. And the other one is 10 decimetres by 10 decimetres. The area of our square, measured in decimetres, would be equal to 10 times 10. That’s the length times the width. So this would be equal to 100 square decimetres. If we look at our diagram, to go from metres to decimetres, we multiply by 10. So to go from square metres to square decimetres, we square the length conversion, which means we multiply by 10 squared, which is the same as multiplying by 100. And to go in the opposite direction from square decimetres to square metres, we would divide by 10 squared, which is the same as dividing by 100.

So now, we know that one square metre is equal to 100 square decimetres. We can add in the following metric area conversions that one square centimetre is equal to 100 square millimetres. One square decimeter is equal to 100 square centimetres. And one square kilometre is equal to 1000000 square metres. However, if we remember the trick of squaring the length conversion, then we can use that alongside the metric length conversions that we already know.

So let’s now have a look at some examples of converting metric areas.

460000 square centimetres equals what square metres.

Let’s start by reminding ourselves of the length conversion that one metre is equal to 100 centimetres. If we took a square of one metre by one metre, the area can be found by multiplying the length by the width to give us one times one, which is one square metre. An equivalent square of 100 centimetres by 100 centimetres will give us an area of 10000 square centimetres. We can think of this if we wanted to change from metres to centimetres, we multiply it by 100. And we can use the rule that, to convert an area, we square the length conversion.

Here, that means if we want to convert between square metres and square centimetres, we would multiply by 100 squared, which is 10000. And to change from square centimetres to square metres, we divide by 100 squared. So now, we can say that one square metre is equal to 10000 square centimetres. So to change 460000 square centimetres into square metres, we take our value of 460000. And we divide by 100 squared. This is the same as 460000 divided by 10000, which is 46 square metres. So the missing answer in the question is 46.

25000000 square metres equals what square kilometres.

Let’s start this by recalling that one kilometre is equal to 1000 metres. This means that if we wanted to change from kilometres into metres, we would multiply by 1000. And if we wanted to change from metres to kilometres, we would divide by 1000. So if we wanted to convert the area units of square kilometres to square metres, we can use the technique that we square the length conversion. This would mean that we multiply by 1000 squared. And if we wanted to go backwards from square metres to square kilometres, we would divide by 1000 squared. And since 1000 squared is equal to 1000000, that means, to convert backwards or forwards, we would multiply by 1000000 or divide by 1000000. We can also say that one square kilometre is equal to one million square metres.

So to answer our question, we have 25000000 square metres. And we want to change it into square kilometres, which means we must divide by 1000 squared or divide by 1000000. And since 25000000 divided by 1000000 is 25, this means that our missing answer for our square kilometres is 25.

Use less than, equals, or greater than to fill in the blank. Seven square metres what 700 square decimetres.

In this question, we’re comparing areas. We know that these are areas because the units are squared. However, we can’t compare seven and 700 directly since the square units are different. Let’s start by writing down a length conversion that we already know that one metre is equal to 10 decimetres. This means that, to change from metres to decimetres, we would multiply by 10. If we want to convert an area, we can square the length conversion. This means that if we want to change square metres into square decimetres, we would simply multiply by 10 squared, which is the same as multiplying by 100. This means that one square metre is equal to 100 square decimetres.

So let’s take the seven square metres in our question and change that into square decimeters. To do this, we would take our value of seven and multiply it by 100, giving us 700 square decimetres. So if we look at the values in our question, seven square metres and 700 square decimetres. Well, we’ve just shown that seven square metres is equal to 700 square decimetres. So the missing symbol in the question must be equals.

Let’s now look at a question where we have to order areas given in different metric units.

Arrange the areas in descending order. 10 square decimetres, 2500 square millimetres, 150 square centimetres.

In this question, we can start by just comparing the values of 10, 2500, and 150 because they all have different units. We have square decimetres, square millimetres, and square centimetres. To order and compare these values, we would have to make all the units the same. We can choose any unit. But since we can easily change square decimetres to square centimetres and we can change square millimetres to square centimetres, then let’s convert them all to square centimetres. We can use the fact that one square decimetre is equal to 100 square centimetres. And that one square centimetre is equal to 100 square millimetres.

So let’s start by changing our 10 square decimetres into square centimetres. If one square decimetre is equal to 100 square centimetres, this means that, to get to our value of 10, that’s like multiplying by 10. So we can multiply the value of 100 by 10 as well. And since 100 times 10 will give us 1000, then 10 square decimetres is equal to 1000 square centimetres. Next, let’s change two and a half thousand square millimetres into square centimetres. Since we know that one square centimetre is equal to 100 square millimetres, that means that if we want to change square centimetres to square millimetres, we would multiply by 100. If we want to go the opposite direction from square millimetres to square centimetres, we would divide by 100. So since we’re changing from square millimetres to square centimetres, we must divide 2500 by 100, which will give us the value 25 square centimetres.

So now, we have three areas given in the same square units. And we need to write them in descending order. That means from the largest to the smallest. Out of our three areas, the value of 1000 square centimetres is our largest one. But instead of writing that first, we must be careful to use the original value of 10 square decimetres. The next largest value will be 150 square centimetres. And finally, our smallest value would be 25 square centimetres, which we write in the form with its original unit of 2500 square millimetres. And so we have the final answer for the areas in descending order.

In our final question, we’ll look at two areas, one given as a fraction. And we need to express the areas as a ratio.

Area a is two-fifths square metres. Area b is 415 square decimetres. Express the ratio of areas a to b in its simplest form.

Let’s start by noticing that the areas are given in different units. The area of a is given in square metres. And the area of b is given in square decimetres. We will need to convert these into the same unit. But let’s start by looking at area a. It will be good if we could write the fractional value of two-fifths as a decimal. We can recall that one-fifth is equal to 0.2. So two-fifths must be equivalent to double that, which is 0.4. So two-fifths of a square metre is equal to 0.4 square metres. Area b is given as 415 square decimetres. So we will need to use a conversion between square metres and square decimetres. And that is that one square metre is equal to 100 square decimetres.

So let’s change our area a in square metres into square decimetres. To do this, we take our value of 0.4. And we multiply it by 100, giving us 40. So now, we can say that area a is equivalent to 40 square decimetres. The question then asks us to express these areas as a ratio, with a first and then b. So we write our values of 40 to 415. And we don’t worry about units in a ratio. Now, we need to see if we can write this ratio in a simpler form. We can notice that since 40 ends in zero and 415 ends in five, then both of these numbers must be divisible by five. So dividing both sides of our ratio by five will give us eight to 83. Since we can’t simplify this any further, this will be our final answer.

So let’s now look at what we’ve learned in this video. We knew the metric length conversions between centimetres and millimetres or decimetres and metres, for example. We have also now learned that there are different values for converting between the area units. And we can find these conversion values by squaring the length conversion. For example, we know that, to change from kilometres to metres, we multiply by 1000. So to change from square kilometres to square metres, we would multiply by 1000 squared, which is the same as multiplying by 1000000. So here, we have our metric area conversions. And it can be very helpful to make a note of these so that we can use them in our calculations.