In this video, we’ll learn how to find the areas of rectangles and squares using a formula. And we’ll learn how to apply this to real-life problems. We’ll begin by recalling some of the key terms that we’ll use in this video, before looking at how counting squares in a rectangle can help us to drive a formula for their area. We’ll then apply this formula to a number of examples, including those with integer and noninteger dimensions, before considering how the formula can help us to calculate missing lengths given the area.
Remember, the area of a shape is the amount of two-dimensional space the shape takes up. We measure it in square units, such as square centimeters, square meters, and square feet. The perimeter, on the other hand, is the total distance around the edges of the shape. And it’s measured in standard units, such as centimeters, meters, or feet.
Now, we can calculate the area of a shape by simply counting square units. For example, let’s take this rectangle. It contains a number of unit squares. These are squares whose side length are one unit each. We can count the number of squares it contains. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18. Since it contains a total of 18 squares, its area is 18 square units. And if we let the dimensions be in centimeters, then we can say that the area of the rectangle is 18 square centimeters. But is there a quicker way to calculate this?
Well, yes, we can see that the rectangle contains three rows of squares. Each row contains six squares. We can therefore calculate the total number of squares and therefore the area of our shape by multiplying three by six, which once again gives us an area of 18 square centimeters. Notice also that we could have alternatively said that the shape contains six columns, each of which contains three squares. This time, we multiply six by three. But since multiplication is commutative, that is, it can be done in any order, we’ll get the same answer.
And in other words, we can say that we can find the area of our rectangle by calculating the product of its dimensions. If we say that the dimensions of our rectangle are its width and its height, then its area is calculated by multiplying its width by its height. This might alternatively be written as width times length or base times height. Now, since a square is simply a rectangle whose dimensions are equal, we can amend the formula slightly and say that the area of a square is the length of its base or one of its edges squared.
Let’s now have a look at the application of this formula.
Emma is buying a new house. She needs to find the area of each room to check whether her new furniture will fit into each of the rooms. The rooms’ dimensions are measured in feet. Find the area of the living room.
We’ve been given the floor plan of a house, and we’re being asked to find the area of the living room. Well, we see that the living room is this space here. And it’s in the shape of a rectangle. So we recall that for a rectangle whose dimensions are labeled width and height, its area is width multiplied by height. In this case, the dimensions are measured in feet. So we’ll define the width of our living room as 11 feet and its height as nine feet.
Now, of course, we know that we’re going to be multiplying these values. And multiplication is commutative. So we could have defined the height as being 11 feet and the width as being nine feet, and we would’ve achieved the same answer. The area of our living room is 11 multiplied by nine, which is 99. And of course, since we’re working with area, we know we’ll have square units, and the units here are feet. So we find the area of the living room to be equal to 99 square feet.
In our next example, we’ll have a look at a real-life application of the area of a square.
This is a quilt square where each side is 4.75 inches. Scarlett plans to use 12 squares identical to the one below to make a small blanket. In square inches, what will be the size of the blanket?
Let’s begin by adding the dimensions to our diagram. We’re told that the side length of our square is 4.75 inches. And of course, we know that squares have four equal sides. 12 such squares, we call these congruent squares, are going to be used to make a small blanket. And our job is to calculate the size of the blanket. In other words, we’re going to be calculating the area of the blanket.
So there are two ways we can do this. We could try and put the 12 squares together to form a blanket and calculate the width and the length. A much more sensible method though is to begin by calculating the area of the square and then multiplying this by 12. And we know that the area of a square is found by squaring one of its side lengths. So we’ll calculate the area of one of our squares.
Now, the side length of our square is 4.75 inches, so its area will be 4.75 squared. And how do we calculate 4.75 squared? Well, as is often the way in mathematics, there is more than one technique. We could convert 4.75 into an improper fraction and go from there. Alternatively, we’ll calculate 475 squared. That’s 475 times 475. Let’s calculate this using the grid method. Four times four is 16, so 400 times 400 is 160,000. Seven times four is 28, so 70 times 400 is 28000. We also know that four times five is 20, so 400 times five is 2000. And we continue filling in our grid by multiplying the number at the top by the number on the side.
Our final step is to add all of these values together. And when we do, we get 225,625. So that’s the value of 475 squared. Of course, we wanted to work out 4.75 squared. That’s 4.75 times 4.75. 4.75 is 100 times smaller than 475. So to find 4.75 squared, we can divide 225,625 by 100 and then by 100 again. Alternatively, we divide it by 10000. That gives us 22.5625, which means the area of one of our squares is 22.5625 square inches.
Now, a nice little way that we can check the accuracy of our answer is to round each digit in our calculation to one significant figure. That means we can estimate 4.75 times 4.75 by working out five times five. That’s 25, which is a little bit bigger than the actual answer we got. So that’s a good indication that we did our calculation correctly.
We know that the area of the blanket is 12 times the area of one of the squares. So we now need to work out 22.5625 times 12. Once again, we can do this by multiplying 225,625 by 12. That gives us 2,707,500. Dividing this value by 10000 gives us the answer to 22.5625 times 12. It’s 270.7500 or just 270.75. And so we find the area of our blanket to be 270.75 square inches.
In our next example, we’ll look at how information about the perimeter of a square can help us calculate its area.
A farmer has 53 and one-third feet of fencing and wants to build a square pen for his chickens? What is the area of the largest pen he can build?
We’re told that the farmer wants to build a square pen using 53 and one-third feet of fencing. Since the fencing encloses the square, we can say that 53 and one-third feet is the perimeter of our square. Now, we also know that squares have four sides of equal length. Let’s call these 𝑥 feet.
So let’s begin by finding the value of 𝑥. To do this, we divide 53 and one-third by four. But we can’t really do that until we’ve converted our mixed number, 53 and one-third, into an improper or top heavy fraction. To do that, we multiply the integer part by the denominator of the fraction. That’s 53 times three, which is 159. We then add this value to the numerator of our fraction. 159 plus one is 160. This part forms the numerator of our top heavy or improper fraction. The denominator stays the same. It’s three.
And so the value of 𝑥 can be calculated by dividing 160 over three by four. Let’s write four as four ones or four over one and remind ourselves that to divide by a fraction we multiply by the reciprocal of that fraction. This process is sometimes called “keep, change, flip.”
We’re now going to cross-cancel by noticing that both 160 and four are divisible by four. When we divide 160 by four, we get 40. 40 times one is 40, and three times one is three. So we find that 𝑥 is equal to 40 over three or forty thirds.
Now, at this stage, we might be tempted to convert this back into a mixed number. But let’s check we’ve actually finished working with our fractions. We now know that the side length of our square is forty thirds feet. And the question asked us to find the area of the largest pen he can build. So we recall that the area of a square is found by squaring its side lengths. So the area of the pen is forty thirds squared.
Notice that had we converted this back into a mixed number, we wouldn’t be able to easily calculate this. But to square a single fraction, we square both the numerator and denominator. 40 squared is 1600, and three squared is nine. So the area is 1600 over nine square feet.
We’re now ready to convert this back into a mixed number. This time, we divide the numerator by the denominator. 1600 divided by nine is 177 with a remainder of seven. 177 forms the integer part of our mixed number, whereas the remainder forms the numerator of the fractional part. The denominator remains unchanged. This means that 1600 over nine is the same as 177 and seven-ninths. And since we’re working in feet, we can say that the area of the pen will be 177 and seven-ninths square feet.
We’ll now consider how being given the information about the area of a rectangle can help us to calculate its side length.
Find the width of a rectangle whose area is 104 square centimeters and length is 13 centimeters.
We’ve been given some information about the area and length of a rectangle. Let’s sketch this out. Next, we need to recall the formula for the area of a rectangle. For a rectangle whose two dimensions are width and length, its area is width multiplied by length. So how do we calculate the width, which is what we’re looking to do here, if we already know the area?
Well, we can use a bit of logic or replace our various dimensions in the formula to create an equation. Let’s let the width of our rectangle be equal to 𝑤 centimeters. We were given that its area is 104 square centimeters. We defined its width to be equal to 𝑤. And we’re told its length is 13 centimeters. So we have 104 equals 𝑤 times 13. In other words, 104 equals 13𝑤.
We’re trying to calculate 𝑤. So we solve this equation by using inverse operations. The 𝑤 is being multiplied by 13. So we divide both sides of our equation by 13. 𝑤 is therefore equal to 104 over 13. 104 divided by 13 though is eight. And so we can say that the width of the rectangle whose area is 104 square centimeters and whose length is 13 centimeters is eight centimeters.
We’ll consider one final problem-solving example.
If the sum of the perimeters of two squares is 112 centimeters and the side length of one of them is nine centimeters, find the sum of their areas.
In this question, we’re given some information about two different squares. Let’s call them 𝐴 and 𝐵. We’re told that the side length of one of these squares is nine centimeters. We’ll assign these lengths to square 𝐵. Now, we’re actually told that the sum of the perimeters of our two squares is 112 centimeters. And we’re looking to calculate the areas.
Now, the formula that we use to calculate the area of a square is side length squared. So we can quite easily calculate the area of square 𝐵. But we can’t calculate the area of square 𝐴 without knowing its side length. So we’re going to begin by calculating the perimeter of square 𝐵 and subtracting this from the total perimeter. Now, the perimeter is the total distance around the square. So the perimeter of square 𝐵 must be nine multiplied by four, since it has four equal sides. Nine times four is 36. So we find the perimeter of square 𝐵 is 36 centimeters.
We know that the sum of the perimeters of our squares is 112 centimeters. So if we subtract 36 from 112, that gives us the perimeter of square 𝐴. That’s 76 centimeters. We said that to calculate the area of square 𝐴, we need to find its side length. Now, remember, when we knew the side length, we calculated the perimeter by multiplying this by four. This time, we know the perimeter and we’re looking to find the side length. So we’re going to divide 76 by four. That’s 19. And so we’ve found that the side length of square 𝐴 is 19 centimeters.
We’re now ready to find the area of each of our squares. The area of square 𝐴 is 19 times 19 or 19 squared. That’s 361. And since we’re working in centimeters, it’s 361 centimeters squared. The area of square 𝐵 is nine squared. That’s 81 square centimeters. The question wants us to find the sum of the areas of our squares. So that’s the sum of 361 and 81, which is 442 or 442 square centimeters.
In this video, we’ve learned that the area of a rectangle — remember, that’s the amount of two-dimensional space the shape takes up — is calculated by finding the product of its width and its height. We sometimes call this width times length or base times height. We saw that since a square is a rectangle whose side lengths are equal, the area of a square is its side length squared. And we learned that these are measured in square units, such as square centimeters and square meters.