lesson
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If an 8-inch personal pizza costs 8,howmuchshoulda16−inchpizzacost?Mostpeopleguess16 because the diameter doubled — but they end up severely undercharged.
When you stretch the lengths of any 2D shape, the boundary and the interior respond in completely different ways.
The Scaling Factor
A scale factor (k) is the number you multiply every linear dimension by to enlarge or reduce a figure.
A linear dimension is any 1-dimensional length on a shape, including side lengths, heights, diagonals, radii, and perimeters.
📊An interactive comparison card showing a 1x1 unit square next to a scaled 2x2 square (scale factor k = 2). The 1x1 square is labeled 'Side = 1, Perimeter = 4, Area = 1'. The 2x2 square is split into a 2x2 grid of 4 colored unit squares, labeled 'Side = 2 (x2), Perimeter = 8 (x2), Area = 4 (x4)'. Uses blue accent for perimeter borders and light orange fills for area blocks on a clean off-white container.
Why does multiplying every side by k double the perimeter, but quadruple the area?
Scaling Perimeter
Because perimeter is simply the total linear distance around a shape, it scales directly by k.
For a rectangle with sides a and b, the original perimeter is P=2a+2b. When each side is multiplied by k, the new perimeter is P′=2(ka)+2(kb)=k(2a+2b)=kP.
Let's test this with a concrete shape: a garden measuring 3 meters by 5 meters.
The original perimeter is 2(3)+2(5)=16 m. If you scale the garden by k=4, the new sides are 12 m and 20 m, giving a new perimeter of 2(12)+2(20)=64 m, which is exactly 16×4.
What happens when we measure the flat surface inside the boundary instead of just the fence around it?