3-D shapes (age 11+)

3D shapes are described by faces (flat or curved surfaces), edges (lines where faces meet) and vertices (corner points). A triangular prism has 5 faces, 9 edges and 6 vertices. A square-based pyramid also has 5 faces but only 8 edges and 5 vertices, so counting sorts them out. Counting needs a system: faces first, then edges around each face, then vertices. Curved shapes break the pattern (a cylinder has no vertices), which is itself a useful test. If two shapes share the same counts, look at face shapes to split them.

Solid geometry studies cubes, prisms, pyramids, cylinders, cones and spheres. Prisms have two matching polygonal ends joined by rectangles; pyramids have one polygonal base with triangles meeting at an apex. Cylinders and cones swap polygons for circles. Cross-sections reveal insides: slicing a cylinder horizontally gives a circle, while slicing a cone vertically through its tip gives a triangle. Nets unfold the surfaces flat: two triangles plus three rectangles folds into a triangular prism.

A prism's matching ends give it its name: triangular for triangle ends, rectangular for rectangle ends. A pyramid takes its name from its base. A square-based pyramid has one square and four triangular faces joining at a single tip. For a right triangular prism, a net must contain two triangles plus three rectangles. Count the faces, then fold a paper net and tape the joins. Faces that overlap or leave holes show that the net does not fit.

Surface area adds every face, and slant faces of pyramids need their triangle areas. A square-based pyramid with base 4 by 4 and four triangles of area 6 has surface 16 + 24 = 40. Prisms add two ends plus the rectangular band around the middle. Building from nets makes area concrete: cut, fold, then add the flat pieces. If the computed faces do not match the solid's faces, the net is wrong before any area work starts.