Question

Ginger had a cardboard cut out of a triangle and decided to cut a piece off of the triangle.

When she did this is the quadrilateral formed of greater of less perimeter? Is the perimeter always less when you cut off a portion of a shape? In general what happens to area when you cut off a piece of a geometric shape?

Answer 1

Short answers first:
- If Ginger cuts off a corner of a triangle by making a straight cut between two points on its sides, the resulting quadrilateral has smaller perimeter.
- Area always decreases (unless the piece you remove has zero area).
- Perimeter can either decrease, stay the same, or increase, depending on how you cut.

Why the triangle → quadrilateral perimeter is smaller (simple proof):
Label triangle ABC. Pick points A1 on AB and A2 on AC and cut along the straight segment A1A2, removing triangle AA1A2. Original perimeter = AB + BC + CA = (A1B + AA1) + BC + (CA2 + AA2). New perimeter = A1B + BC + CA2 + A1A2. So the change is
new − old = A1A2 − (AA1 + AA2).
But in triangle AA1A2 the straight side A1A2 is shorter than the two other sides: A1A2 < AA1 + AA2. Hence new − old < 0, so the perimeter decreased.

General rules about area and perimeter:
- Area: removing any nonzero-area piece always reduces the area.
- Perimeter: compare the length of the cut you add to the length of boundary you removed.
- If you replace a boundary arc (or two boundary segments) by a straight chord joining its endpoints, the chord is always no longer than the arc or broken path it replaces, so the perimeter decreases.
- If you cut out an interior piece (create a hole), you add the new inner boundary, so the total perimeter increases.
- If the cut is a very long wiggly curve that removes a short portion of original boundary, the new cut length can exceed the removed length, so the perimeter can increase.
- Equality (no net change) is only in degenerate cases when the length of added cut exactly equals the length of removed boundary.

Examples to keep in mind:
- Cutting a corner off a polygon by a straight line lowers perimeter.
- Punching a hole in a sheet (removing an interior disk) increases total boundary length.

So for Ginger’s cardboard triangle (cutting off a corner with a straight cut), the quadrilateral has strictly smaller perimeter and smaller area.