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