Folded-corner perimeter
See why a right-angled notch at the corner of a rectangle does not change the perimeter: inner steps replace exactly what was removed from the top and right.
Drag any of the four original rectangle corners. Each corner can fold inward independently, and you can fully unfold back to a plain rectangle.
TL(0,0) TR(4,3) BR(0,0) BL(0,0)
P = 2W + 2H = 2×12 + 2×10 = 44
Each fold removes some outer lengths, but adds exactly the same lengths inside the shape, so the total perimeter is unchanged.
Drag any of the four original rectangle corners. Each corner can fold inward independently, and you can fully unfold back to a plain rectangle.
TL(0,0) TR(4,3) BR(0,0) BL(0,0)
P = 2W + 2H = 2×12 + 2×10 = 44
Each fold removes some outer lengths, but adds exactly the same lengths inside the shape, so the total perimeter is unchanged.
Folded-corner perimeter
Start with a rectangle of width W and height H. Its perimeter is 2W + 2H.
Cut out a right-angled notch from the top-right corner: go in a units along the top edge, then down b units along the right edge (the same idea as folding that corner inward). The boundary becomes a rectilinear L-shape.
Why the perimeter does not change
Walk around the new shape in order. You still trace the full bottom (length W) and the full left side (length H). Along the top you only walk W - a, and along the outer right you only walk H - b — but inside the notch you add a horizontal step of length a and a vertical step of length b. Altogether:
W + (H - b) + a + b + (W - a) + H = 2W + 2H
So the perimeter matches the original rectangle no matter how you choose a and b (as long as the notch stays inside the rectangle).
Use the interactive panel at the top to drag the sliders and turn on Show why it stays constant to see the a and b segments matched with the “missing” parts of the top and right.
Practice
For printable composite perimeter questions, open the Counting Perimeter worksheet.