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
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.