Surface Area of Solids

Objective: Add the six face areas of a closed box.

Prerequisite skills: areas of rectangles and circles. A closed rectangular prism has two faces of each pairwise dimension product. Surface area totals those faces rather than multiplying all three dimensions as volume does. Common error: using volume lwh as surface area. Restrictions and interpretation: Surface area is an editorial supporting extension; include only exposed faces and state whether ends or tops are present.

Objective: Distinguish curved surfaces from circular bases.

Unrolling a cylinder’s lateral surface forms a rectangle of width 2πr and height h. A closed cylinder adds two disks πr²; an open-top cylinder adds one. A right circular cone has lateral area πrℓ, where ℓ is its slant height, and adds one base disk when closed. A sphere has surface area 4πr² and no separate base. These surface formulas use square units and differ from volume formulas. Common error: using volume in place of surface area or using perpendicular height in a cone’s slant-height formula. Restrictions and interpretation: Surface area is an editorial supporting extension; include only exposed faces and state whether ends or tops are present.

Objective: Subtract hidden shared faces from a sum of surface areas.

When two solids are joined along a complete common face, that face is internal on both solids. Add their separate surface areas and subtract twice the shared-face area. The shared face is counted twice before joining. Common error: subtracting the shared face only once. Restrictions and interpretation: Surface area is an editorial supporting extension; include only exposed faces and state whether ends or tops are present. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.