Weird shape area

Some ACT Math area questions will involve shapes that are irregular, meaning that they are not the standard triangle, circle, rectangle, or square. Instead, they might simply be polygons that combine portions of these shapes together.

• For weird shaped area problems, break down the large, weird shape into smaller, more familiar shapes. Just like with equations, it’s about making the unfamiliar seem more familiar.

Sample Problem

Here is an example problem from the July 2019 ACT:

There are two different ways to approach this problem. Either, the area of the original rectangle can be calculated, as well as the squares that were removed, and then the two numbers can be subtracted. Or the remaining shape can be broken down into a group of five rectangles, and their areas can be summed up together.

Now, since the answers to the question all involve subtraction, it would make sense that the subtraction method might work best in this case.

The original area of the rectangle was 18 • 8 = 144. The area of each square removed was equal to x². Four of these squares were removed, and so the area of the remaining rectangle is 144 − 4x² or G.

Key Takeaways

With weird shape areas it’s often helpful to remember that weird shapes are made from cutting out familiar shapes from familiar shapes, or by adding familiar shapes to familiar shapes. 

Also remember that especially when it comes to shapes made from cutting things out. There are some quick rules to remember about how this affects the areas and perimeters of the shapes.

• Area: Just like the question above, remember that the area of the larger shape can still be calculated to find the area of the new shape. It’s just about remembering to take away the areas of the shapes that were cut out.
• Perimeter: A slight more fun trick. If the shapes that are cut out from the original shape are rectangles (are made up of only perpendicular lines), then the perimeter of the new shape is the same as the perimeter of the old shape. So in the above example, the original rectangle and the new shape would have the same perimeter.

The perimeter trick is particularly useful for saving time on questions. Perimeter doesn’t change by taking rectangles out of shapes.