When You Cut Off A Piece From A 2D Shape, Does The Area And Perimeter Decrease, Increase Or Stay The Same?

Area and Perimeter are two basic properties of 2-Dimensional shapes, and is often the first introduction to geometry in early elementary levels. It is important to have an intuitive understanding of how these two properties varies when the shape changes. For example, what happens to the Area and Perimeter when you cut off a piece from a 2D Shape?

Area

The effect on area when you cut off a piece from a 2D Shape is straightforward. Imagine holding a square piece of paper and cutting off one corner with a pair of scissors. The piece that you’ve just cut off fell on the table. No matter what shape your scissor blades are, the piece on the table always have some area. And since you did not add more paper to the situation, the “area” of the cut piece must have been “taken” from the original square you held in your other hand. Hence the area must be reduced when you cut off a piece from a 2D Shape.

Perimeter

Now, the situation with perimeter is more interesting. It depends on how the shape is cut.

Perimeter Reduces (Less)

Back to our square paper example. If you would just do a simple straight cut off a corner, i.e. cut off a small triangle, the situation would look like this:

Fig 1 – Cut Shape Perimeter Reduces

In the figure above, the blue line represents the perimeter of the original shape, the yellow segment is removed from the original perimeter and the red is added. Since the red segment is shorter than the yellow segment, we’re taking away more than we add, Hence, the perimeter is less than the original.

Perimeter Remains the Same (Constant)

Now imagine you’d cut a square off the corner, like this:

Fig 2a – Cut Shape Perimeter Stays The Same

Here, the length of the red segment is exactly the same as the yellow segment. That is. we remove and add the same length to the perimeter. Hence the perimeter stays the same.

Furthermore, if you keep cutting right angles, you’d end up in a staircase shape edges.

Fig 2b – Cut Shape Perimeter Stays The Same
Fig 2c – Cut Shape Perimeter Stays The Same

You guess it! The perimeter stays the same in all these cases, because the length of the red segment is always the same as the yellow segment.

Perimeter Increases (More)

In that case, can perimeter increase when you cut off a piece from a 2D Shape? Well, yes! Here’s an interesting cut where the red segment is longer than the yellow segment.

Fig 3a – Cut Shape Perimeter Increases

You can also, just cut a wavy line along what would have been the right angle edge in Fig 2a.

Fig 3a – Cut Shape Perimeter Increases

The wavy lines are always longer than the straight lines. So, the red segment is longer than the yellow segment again.

Conclusion

As you can see from the above examples. While area always reduces or become less when you cut off a piece from a 2D Shape, perimeter can reduce (become less), stay the same (constant) or even increase (become more)!

How about another interesting question on perimeter?

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