A rectangle with perimeter 28 millimeters is shown in the diagram. Find the proportion of the shaded area inside the rectangle.
The word “proportion” is telling us to find the shaded area as a fraction of the whole shape. To do this then, we’ll need to calculate the area of the whole shape and the area of the shaded region. To find the area of the shaded region, we’ll first find the area of the rectangle and then we’ll subtract the area of the two triangles.
Notice how we’re told that the perimeter of the shape is 28 millimeters. We can use this then to help us find the width of the rectangle. We know that in a rectangle, opposite sides have the same length. So we can start by subtracting two lots of six from the perimeter of 28 millimeters. That gives us 16 millimeters.
16 millimeters represents two lots of the width of the rectangle. We can halve that then to get that the width of the rectangle is eight millimeters. Once we have all these measurements, we can work out the missing dimensions of the two triangles.
The missing height of this triangle at the top of our diagram is calculated by subtracting four from six. Six minus four is two. So it has a height of two millimeters. Similarly, the missing width of this triangle to the left of our diagram is calculated by subtracting six from eight. It’s also two millimeters.
The formula for the area of a rectangle is its width multiplied by its height. We calculated earlier that the width of this rectangle is eight millimeters and it has a height of six millimeters. Eight multiplied by six means it has an area of 48 millimeters squared.
Next, we need to find the area of the two triangles. The formula for area of a triangle is a half multiplied by its width and multiplied by its height. In the case of this triangle on the left then, that’s a half multiplied by two multiplied by six. And a half of two is one. So this becomes one multiplied by six, which is six. This triangle has an area of six millimeters squared.
For the triangle at the top of the diagram, its area is a half multiplied by eight multiplied by two. A half of eight is four and four multiplied by two is eight. So this area is eight millimeters squared.
Remember we said that to find the shaded area, we’d need to subtract the area of the two triangles from the area of the rectangle. 48 minus six plus eight is equal to 34 millimeters squared. Finally, to write this as a proportion, we have to write as a fraction of the whole shape. We said that the whole shape, which is a rectangle, has an area of 48 millimeters squared. So the proportion is given by 34 over 48. We can simplify this fraction by dividing both the numerator and the denominator by two.
The proportion of the shaded area inside the rectangle is 17 out of 24.