The sum of the perimeters of a rectangle and a square is 138 centimeters. The width of the rectangle and the length of the square both equal 16 centimeters. Find the perimeter and the area of the rectangle.
In this question, there are two shapes, a rectangle and a square. We’re told the sum of the perimeter of the rectangle and the square and we’re asked to find the perimeter and area of the rectangle. Let’s start by reminding ourselves what the perimeter and the area is. The perimeter of a shape is the distance around the outside of the shape. The area is the amount of space is taken up by the shape. So let’s draw our sketch of the square and the rectangle.
We’re told that the width of the rectangle and the length of the square both equal 16 centimeters. And for our square, since we have four sides of equal length, this means that all the sides of our square will be equal to 16 centimeters. We’re also told that the sum of the perimeters of the rectangle and the square is 138 centimeters. This means that if we add together all of the edges of the square and all of the edges of the rectangle together, we’d get 138 centimeters. Let’s see if we can like this in a more formal mathematical way by writing an equation. Starting with the perimeter of the square, since we know that the perimeter is the distance all around the outside. We can write this as 16 plus 16 plus 16 plus 16 or simply four times 16, which we can evaluate as 64 centimeters.
For the perimeter of the rectangle, we know that the width is 16 centimeters. We don’t know the length, but let’s create a value 𝑥 which will represent the length. So to find the perimeter, we could write this as 𝑥 plus 𝑥 plus 16 plus 16 which is equal to two 𝑥 plus 32. We can’t do anything more with the perimeter of the rectangle from here. But let’s return to the fact that we were told the sum of the parameters of the rectangle and the square is 138 centimeters. This means that we can create an equation for the sum of the perimeters as 64 plus two 𝑥 plus 32 and put that equal to 138. Collecting our numerical values of 64 and 32 will give us two 𝑥 plus 96 equals 138. Then to rearrange our equation to get 𝑥 by itself, we subtract 96 from both sides of our equation, giving us two 𝑥 equals 42. And then dividing by two to find 𝑥 will give us 𝑥 equals 21 centimeters.
And so we find out that the length of our rectangle is 21 centimeters. So to find the perimeter of our rectangle, we would calculate 21 plus 16 plus 16. Or since we’ve already worked out that the perimeter of the rectangle is equal to two 𝑥 plus 32, now we know 𝑥 is 21. We can substitute in that value. Using either method, we would find that the perimeter of the rectangle is 74 centimeters. To find the area of the rectangle, we use the formula that the area of a rectangle is equal to length times width. Since we know that our length is 21 and our width is 16, this will give us the area equals 21 times 16, which we can evaluate as 336. The units here will be square centimeters since we’re dealing with an area. So our final answer is the perimeter is 74 centimeters and the area is 336 square centimeters.