Area and perimeter
A flat shape has an edge and an inside. How do you measure each one, and why does knowing one not tell you the other?
Where you are 15 inside
What this area is about
Perimeter is the length all the way around a flat shape, counted in length units. Area is how much flat surface it covers, counted in square units. This area builds both, then finds the area of shapes that have no formula of their own.
How these fit together
What area and perimeter are. Area as covering, measuring with unit squares and splitting a rectangle into rows and columns of squares set up area. Perimeter adds the sides. Perimeter versus area shows the two are separate measures.
Formulas. Rectangles, parallelograms and trapezoids, triangles and regular polygons each get a formula and the reason for it. Why the circle formula works explains A = \pi r^2.
Procedures. Splitting into rectangles, cutting any polygon into pieces and composite figures cut a shape into pieces with formulas, or take a known piece away from a bigger shape. A missing side from the area and from the perimeter run the formulas backward.
A path through it
Area: covering, then unit squares (splitting a rectangle into squares can sit beside them), then rectangles. From rectangles, splitting into rectangles, then parallelograms and trapezoids, then triangles, then regular polygons, any polygon and composite figures.
Perimeter: the sum of the sides, then the missing side. Once both are in hand, the missing side from the area (its second half uses perimeter) and perimeter versus area. The circle formula waits for the area of a circle.
Next: volume, surface area and the Pythagorean theorem.