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Grade 6 (age 11-12)

grade 6 Prisms and Pyramids

Prisms and pyramids are solids whose faces are all polygons. A prism has two parallel, congruent bases and rectangular side faces, like a hexagonal pencil or a triangular chocolate box. A pyramid has a single base, and its triangular side faces meet at one point, so it comes to a point like the pyramids of Egypt. Change the number of sides of the base, n, and turn the shape to see how the numbers of vertices, edges and faces depend on n, and what the unfolded net looks like.

What is a solid?

📦 Flat shapes and solids
①A flat shape can be drawn on paper, while a solid, like a box or a pyramid, takes up space
②On a solid, the flat surfaces are faces, the line segments where two faces meet are edges, and the points where edges meet are vertices
③Prisms and pyramids are solids whose faces are all polygons, and they are named after the shape of their base
④With a four-sided base you get a rectangular prism or a square pyramid

Looking at prisms

🏛️ Looking at prisms
①A prism has two parallel, congruent bases, one at the top and one at the bottom
②Its side faces are all rectangles, one for each side of the base
③The starting shape, a rectangular prism, has 8 vertices, 12 edges and 6 faces, the same numbers shown under the picture
④A hexagonal pencil is a hexagonal prism, and a triangular chocolate box is a triangular prism
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Looking at pyramids

↔️ The same n and angle
The number of base sides n and the angle you chose above are used for the pyramid below too.
🔺 Looking at pyramids
①A pyramid has just one base, and its triangular side faces meet at a single point at the top, called the apex
②The side faces are all triangles, one for each side of the base
③The starting shape, a square pyramid, has 5 vertices, 8 edges and 5 faces
④The pyramids of Egypt are square pyramids

Nets

✂️ Looking at nets
①Cutting a solid along its edges and laying it flat gives its net
②The net of an n-sided prism has 2 bases and n rectangular side faces, so n + 2 faces
③The net of an n-sided pyramid has 1 base and n triangular side faces, so n + 1 faces
④Fold the net back up and you get the solid again, so edges that meet must be the same length

Summary

n-sided prism
Vertices = 2n, Edges = 3n, Faces = n + 2
2 bases (n-gons) and n side faces (rectangles)
n-sided pyramid
Vertices = n + 1, Edges = 2n, Faces = n + 1
1 base (an n-gon), n side faces (triangles) and the apex
Prism and pyramid
prism: 2 congruent bases / pyramid: 1 base and an apex
A prism's side faces are rectangles; a pyramid's are triangles
🎯 At a glance
①Prism: two congruent bases, rectangular side faces
②Pyramid: one base, triangular side faces meeting at an apex
③n-sided prism: 2n vertices, 3n edges, n + 2 faces
④n-sided pyramid: n + 1 vertices, 2n edges, n + 1 faces
⑤Folding a net gives back the solid

Examples & Unit Review

Example 1
How many vertices, edges and faces does a pentagonal prism have?
1
An n-sided prism has 2n vertices, 3n edges and n + 2 faces, and a pentagonal prism has n = 5.
2
Vertices 2 × 5 = 10, edges 3 × 5 = 15, faces 5 + 2 = 7.
▸ Vertices 10, edges 15, faces 7
Set n to 5 in the picture and the same numbers appear under it.
Example 2
How many vertices, edges and faces does a hexagonal pyramid have?
1
An n-sided pyramid has n + 1 vertices, 2n edges and n + 1 faces, and a hexagonal pyramid has n = 6.
2
Vertices 6 + 1 = 7, edges 2 × 6 = 12, faces 6 + 1 = 7.
▸ Vertices 7, edges 12, faces 7
A pyramid has the n vertices of its base plus the apex, so n + 1 vertices.
Unit Review
How many edges does a triangular prism have?
①9
②6
③12
④5
▸ ① 9
1
An n-sided prism has 3n edges, and a triangular prism has n = 3.
2
3 × 3 = 9, so it has 9 edges.
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