Solids in space and their parts

A solid is a shape in space: it has a length, a width and a height. These are the course’s seven solids:

1. Box – six faces, all of them rectangles: a shoebox, a room, a shipping container.
2. Cube – a box whose edges are all equal: a die.
3. Prism – two equal, parallel bases (here triangles), and the faces between them are rectangles: a tent, the roof of a shed.
4. Cylinder – two bases that are equal discs, and a curved surface between them: a can, a water tank, a pillar.
5. Pyramid – one base (here a square), and side faces that are triangles meeting at one point, the apex.
6. Cone – a base that is a disc, and a curved surface narrowing to one point: an ice-cream cone, a pile of sand.
7. Sphere – every point on the sphere’s surface is at the same distance $\displaystyle r$ from its centre: a basketball.

In every solid of the course, the line between the centres of the bases (or between the apex and the centre of the base) is perpendicular to the base.

A solid whose faces are all flat – a box, a prism, a pyramid – is made of three kinds of parts:

• Face – a flat surface bounding the solid. A box has 6 faces.
• Edge – a segment where two faces meet. A box has 12 edges.
• Vertex – a point where several edges meet. A box has 8 vertices.

And three words for calculations:

• Base – the face the solid “stands” on. A prism and a cylinder have two bases; a pyramid and a cone have one.
• Lateral surface – everything around, apart from the bases: the side faces, or the curved surface of a cylinder or a cone.
• Height – the distance between the two bases, or from the apex to the base, along a line perpendicular to the base.

The parts of a prism and of a pyramid are counted by the number of sides of the base, $\displaystyle n$:

• Prism: two bases, and one side face for each side of the base. Each base has $\displaystyle n$ edges, and $\displaystyle n$ more edges join the bases. Each base has $\displaystyle n$ vertices.
• Pyramid: one base, and one triangle for each side of the base. The base has $\displaystyle n$ edges, and $\displaystyle n$ more edges rise from it to the apex. The vertices: $\displaystyle n$ in the base, and the apex.

For example, a box is a prism whose base is a rectangle, $\displaystyle {n}={4}$: it has $\displaystyle {4+2}={6}$ faces, $\displaystyle {3\cdot4}={12}$ edges and $\displaystyle {2\cdot4}={8}$ vertices.

Which solid is each object like? Write the solid’s name.

1. A drinks can
2. A die
3. A tent whose front is a triangle
4. An ice-cream cone
5. A football
6. The roof of a tower: a square base, and triangles meeting at the top
7. A lorry’s container

An unsharpened pencil is a prism whose base is a hexagon (6 sides).
How many does it have:

The roof of a garden pavilion is a pyramid whose base is an octagon (8 sides).
Along every edge of the roof there is a wooden beam.
How many wooden beams does the roof have?

A tent is a triangular prism standing on a rectangular face, and it has a floor.
Each face of the tent is one piece of cloth.
A strengthening strip is sewn along each edge.