Chapter 13 · Surface area and volume

Parts of Solids and Their Nets

Learn to name prisms and pyramids by their bases, count their faces, edges and vertices, and match each face of a solid to one shape of its net.

15 min read

A three-dimensional figure, or solid, is a shape that takes up space, like a box. The solids in this lesson have only flat surfaces, and you can describe each one by its parts:

  • A face is a flat surface of the solid. Each face is a polygon, such as a rectangle or a triangle.
  • An edge is a line segment where two faces meet.
  • A vertex (plural vertices) is a corner point where edges meet.

A drawing shows a solid from one side, so some of its parts are out of sight at the back. Edges that are out of sight are hidden edges, and this lesson's drawings show them as dashed lines. The drawings are not to scale, so never measure them: use the lengths stated in the text. Lengths are in units, and areas are in square units.

A rectangular prism is shown. Each of the edges of solid A is described separately below. The edge where the top face meets the front face. The edge where the top face meets the back face. The edge where the top face meets the left face. The edge where the top face meets the right face. The edge where the bottom face meets the front face. The edge where the bottom face meets the back face. The edge where the bottom face meets the left face. The edge where the bottom face meets the right face. The edge where the front face meets the left face. The edge where the front face meets the right face. The edge where the back face meets the left face. The edge where the back face meets the right face.
Solid AA is shaped like a box; its dashed lines are hidden edges, which still count among its 1212 edges.

Prisms and pyramids are named by their bases, special faces you can find this way:

  • A prism has two bases that match: they are the same shape and size, and they face each other. Every prism in this lesson stands straight: each edge that joins its two bases meets them at right angles, so the faces between the bases are rectangles, one for each side of a base.
  • A pyramid has one base. A triangle rises from each side of the base, and all the triangles meet at one vertex, called the apex.

The shape of the base completes the name. Solid AA is a rectangular prism: all 6 of its faces are rectangles and opposite faces match, so any two opposite faces can serve as its bases. A triangular prism has triangle bases, and a square pyramid has a square base. A base is not always the face that a solid rests on, so look for the matching pair of faces, or for the face across from the apex.

A triangular prism is shown. Each of the faces of solid B is described separately below. The front triangular face. The back triangular face. The bottom face. The left slanted face. The right slanted face.
Solid BB has two matching triangles joined by rectangles, and its hidden edges are dashed.

Worked example

Naming and counting a prism

Name solid BB and count its faces, edges and vertices, including the hidden ones.

  1. Find the bases. Solid BB has two matching triangles that face each other, joined by rectangles. Two matching bases joined by rectangles make a prism, and these bases are triangles, so solid BB is a triangular prism.

  2. Count the faces: the 2 triangle bases, plus one rectangle for each of the 3 sides of a base, because each rectangle joins one side of a base to the matching side of the other.

    2+3=52 + 3 = 5

  3. Count the edges. Each triangle base has 3 edges, and one more edge joins each of the 3 corners of one base to the matching corner of the other. Every edge belongs to exactly one of these groups, so none is counted twice and no dashed edge is missed.

    3+3+3=93 + 3 + 3 = 9

  4. Count the vertices. Every vertex is a corner of one of the two bases, and each base has 3 corners.

    3+3=63 + 3 = 6

Answer

Solid BB is a triangular prism with 55 faces, 99 edges and 66 vertices.

A square pyramid is shown. The drawing is not to scale. Labelled elements: solid C. Measures: the base length of solid C; the height of solid C.
Solid CC: each base edge is 66 units, and the marked height, from the apex straight down to the base, is 44 units.

Worked example

Naming and counting a pyramid

Solid CC has a square base with sides of 66 units, and its apex is directly above the center of the base. Its height is 44 units: the distance from the apex straight down to the base, meeting the base at a right angle. Name solid CC and count its faces, edges and vertices, including the hidden ones.

  1. Find the base. Solid CC has one square base, and all its other faces are triangles that meet at the apex, so it is a square pyramid.

  2. Count the faces. A pyramid has one triangle on each side of its base, so there is the square base plus 4 triangles.

    1+4=51 + 4 = 5

  3. Count the edges. The base has 4 edges, and each of its 4 corners is joined to the apex by one more edge.

    4+4=84 + 4 = 8

  4. Count the vertices. Every vertex is either a corner of the base or the apex.

    4+1=54 + 1 = 5

Answer

Solid CC is a square pyramid with 55 faces, 88 edges and 55 vertices.

Key concept

Each face becomes one shape of the net

A net of a prism or a pyramid is a flat pattern of all its faces, joined along edges, that would fold up into the solid with no gaps and no overlaps. Each face of the solid matches exactly one shape of the net, with the same shape and the same size. So the bases tell you what a net holds: for the prisms in this lesson, the 2 bases and one rectangle for each side of a base; for a pyramid, the base and one triangle for each side of the base.

Net of a square pyramid

A net with one square base 6 units on each side and four triangular faces. Each triangle has base 6 units and slant height 5 units.

The net of solid CC: a square with sides of 66 units and 44 triangles whose heights are each 55 units.

Worked example

Matching a pyramid to its net

The net above is the net of solid CC. Match each face of solid CC to one shape of the net. Then use the lengths on the net to find the area of each face and the total area of all 55 faces.

  1. Match the base. Solid CC has one square face, its base, and the net has one square, with sides of 66 units, the same as the base edges of solid CC. So the square is the base, laid flat.

  2. Match the slanted faces, the 4 triangles that slope up to the apex. The apex of solid CC is directly above the center of its base, so these faces are identical triangles, each with a base edge of 66 units. Each one matches one of the 4 triangles on the net, so the net has one shape for each face.

    1+4=5 shapes1 + 4 = 5 \text{ shapes}

  3. Read the height of each triangle. The 55 on the net is the slant height: it runs up a slanted face from the middle of its base edge to the apex, at a right angle to that edge, so it is the triangle's height. It is not the pyramid's height of 44 units, which runs from the apex straight down through the inside of the solid.

  4. Find the area of the square base, which has sides of 66 units.

    6×6=36 square units6 \times 6 = 36 \text{ square units}

  5. Find the area of one triangle from its base edge and its height, the slant height. Then multiply by 4, because the 4 triangles are identical.

    12×6×5=15 square units,4×15=60 square units\tfrac{1}{2} \times 6 \times 5 = 15 \text{ square units}, \quad 4 \times 15 = 60 \text{ square units}

  6. Add the areas. Each face appears exactly once on the net, so this sum is the total area of all the faces of solid CC.

    36+60=96 square units36 + 60 = 96 \text{ square units}

Answer

The square of the net is the base of solid CC, and each of its 4 triangles is one slanted face. The base has an area of 3636 square units, each slanted face 1515 square units, and all 55 faces together 9696 square units.

The same matching works for a prism. The net below belongs to a different triangular prism from solid BB: its bases are right triangles, triangles with a right angle. The two sides that form the right angle are the legs, 33 and 44 units long, and the third side is 55 units. The prism is 88 units long, meaning each edge that joins its bases is 88 units. Like solid BB, it has 2 triangular faces and 3 rectangular faces.

  • The 2 triangles are the bases. Their legs meet at a right angle, so one leg is the height for the other: each base has an area of 12×3×4=6\frac{1}{2} \times 3 \times 4 = 6 square units.
  • The 3 rectangles each join one side of a base to the matching side of the other base, so each is 88 units long and as wide as that side: 3×8=243 \times 8 = 24, 4×8=324 \times 8 = 32 and 5×8=405 \times 8 = 40 square units.

So all 5 faces together have an area of 6+6+24+32+40=1086 + 6 + 24 + 32 + 40 = 108 square units.

Net of a triangular prism

A net with three rectangles, each 8 units long and respectively 4, 3, and 5 units wide, plus two right triangles with perpendicular legs 3 and 4 units.

Two right triangles with legs of 33 and 44 units, and three rectangles 88 units long and 33, 44 and 55 units wide.

Takeaway: Name a prism or a pyramid by its bases, then count its faces, edges and vertices from that structure, hidden ones included. Its net holds each face exactly once, so every shape of the net matches one face of the solid. Next, you will use bases and heights like these to find the volume of prisms and pyramids.

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Volume of Prisms and Pyramids

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