The rectangle shown below has a perimeter of 98 centimeters. What is its area?
We recall that the perimeter of any shape is the distance around the outside. As a rectangle has two equal length pairs of parallel sides, we can calculate the perimeter by adding the length and width and then multiplying by two. The perimeter is equal to two multiplied by six 𝑦 plus seven plus 𝑦. We are told that this is equal to 98 centimeters. Simplifying the terms inside the parentheses or brackets by collecting like terms gives us seven 𝑦 plus seven.
At this stage, we could either distribute the parentheses or divide both sides by two. Dividing the left-hand side by two gives us seven 𝑦 plus seven. 98 divided by two is equal to 49. Our next step is to subtract seven from both sides. This gives us seven 𝑦 is equal to 42. Finally, we divide both sides of this equation by seven. Our value for 𝑦 is six.
We notice on the diagram that our units were centimeters. The width or height of the rectangle is, therefore, six centimeters. Substituting 𝑦 equals six into the expression six 𝑦 plus seven gives us 43. This means that the length of the rectangle is 43 centimeters.
In order to calculate the area of any rectangle, we multiply the length by the width. 43 multiplied by six is equal to 258, as 40 multiplied by six is 240 and three multiplied by six is 18. Adding these gives us 258. If the perimeter of the rectangle is 98 centimeters, then its area is 258 square centimeters. We always measure area in square units.