Surface area
How much material covers a solid exactly once, with nothing left bare and nothing counted twice?
Where you are 3 inside
What this area is about
The surface area of a solid is the total area of its outside, measured in square units such as \text{cm}^2. For a solid with flat faces it is the sum of the face areas, and a net lays those faces flat so that each is counted once. Curved solids need one more idea each: a curved side that unrolls, or a formula of its own.
How these fit together
All three children are procedures, one per family of solids.
Wrapping straight sides. Prisms and cylinders: for a right prism or right cylinder, the two congruent bases plus the side faces, which unroll into one rectangle whose length is the perimeter of the base and whose width is the height.
Narrowing to a point. Pyramids and cones: the base plus the lateral faces, measured along the slant height \ell, not the vertical height. For a regular pyramid or a right cone, the Pythagorean theorem links \ell to the height.
Nothing to unfold. The sphere cannot be flattened without stretching, so it has its own result: 4\pi r^2, four times the area of a circle of the same radius r.
A path through it
None of the three needs another, so any order works: prisms and cylinders, pyramids and cones and the sphere each lean on earlier ideas (nets, circle area, the Pythagorean theorem, the sphere itself), not on one another.
After this, modeling with solids uses surface area to design containers and estimate material.