Faces, edges and vertices
A box and a pyramid feel different in your hands. What exactly is different about them, and how can you say it so someone who cannot see them knows which is which?
Where you are 7 in Three-dimensional shapes
Why this exists
A child who can name solid shapes can point at a box and say “box”, and at a can and say “can”. That is a good start, but a name says nothing about why two solids are different. Ask a child to describe a pyramid to a friend on the phone and the name runs out fast.
Flat shapes already have words for their parts: sides and corners (Defining and non-defining attributes of shapes). A triangle is the shape with 3 straight sides. Solids need the same kind of words, and there are three of them. A face is a flat surface, an edge is a line where two faces meet, and a vertex is a corner where edges meet. With those words, and a count of each, a child can say exactly what makes a cube a cube and a pyramid a pyramid, can tell solids apart in a bag by touch alone, and can check that a solid built from paper or playdough is the one that was meant.
Counting the parts is also harder than it looks. Some faces are underneath and some edges are around the back, so a child who looks once from the front counts too few, and a child who turns the solid around without a plan counts some parts twice. The habit this lesson builds is to count with a plan: top, bottom and around, marking each part once it is counted.
Definition
A solid takes up room: it can be held, and it has an inside you cannot see. Its outside is made of surfaces.
- A face is a flat surface of a solid. It lies flat on a table when the solid sits on it, and pressed into soft playdough it leaves a flat print with the shape of the face.
- An edge is a straight line where two faces meet. Run a finger across a box from one face to the next and the finger goes over an edge.
- A vertex is a point where edges meet: a corner of the solid. The word for more than one is vertices. “Corner” is the everyday word and is fine to use alongside it.
A curved surface, like the side of a can or the outside of a ball, is not flat, so it is not a face in this lesson. The four solids with only flat faces that this lesson counts are these.
- A cube has 6 faces, and every face is a square of the same size. A number cube (a die) is one.
- A rectangular box has 6 faces, and every face is a rectangle. A cereal box and a shoe box are rectangular boxes. Older students call it a rectangular prism. A cube is a special rectangular box: its rectangles are all squares of the same size.
- A square pyramid is a pyramid with a square base: one square on the bottom and four triangles that lean in to meet at a single point at the top.
- A triangular prism has two triangle ends, joined by three rectangles. A tent and a playground ramp have this shape.
Intuition
The lesson runs over three days. Part 1 counts faces, Part 2 counts edges, and Part 3 counts vertices and then names a solid from its counts. Real solids help most: boxes of different sizes, blocks, a can, a ball, and playdough to make the solids and to press faces into.
Part 1: faces
Set a box on the table. A child who counts the faces they can see usually finds 3: the top, the front and one side. The rest are hidden. The plan that finds them all is top, bottom, around. Count the top. Lift the box and count the bottom. Then turn the box a quarter turn at a time and count the faces that go around its middle: for a box, 4 of them. That gives
To be sure no face is counted twice, mark each face once it is counted: a sticky note on a cardboard box, or a small fingertip dent in a playdough solid. When every face has a mark, the count is done.
The faces also have shapes, and the shapes are flat shapes the child knows. Press each face of a playdough solid into a flat slab of playdough, or trace around it on paper, and the print shows the face. A cube’s prints are all squares of one size. A rectangular box’s prints are rectangles, in matching pairs: top and bottom match, front and back match, left and right match. A square pyramid prints one square and four triangles. A triangular prism prints two triangles and three rectangles.
Now try it with the solids in the next picture. A can has a flat circle on the top and another on the bottom, so it has 2 flat faces. Its side is curved: roll it across the playdough and it leaves a long smear, not a shape, because no part of it is flat. A cone has 1 flat face, its circle. A ball has no flat face at all.
Part 2: edges
Run a finger along the line between the top of a box and its front. That line is an edge: it is where two faces meet. A good way to see why an edge belongs to exactly two faces is to fold a sheet of paper. The fold is a line, and there is a flat piece of paper on each side of it. Every edge of a box is a fold like that.
The plan for counting edges is the same as for faces. Count the edges around the top face, then the edges around the bottom face, then the edges that run up from the bottom to the top. On a cube that is 4, then 4, then 4.
A cube has 12 edges. A rectangular box has 12 too, for the same reason: 4 around the top, 4 around the bottom and 4 upright. A strip of tape or a playdough snake laid along each edge is a good way to mark them.
A pyramid is counted the same way with a different plan: the edges around the square base, then the edges that run up to the point. A triangular prism is counted end, other end, then the edges that join the ends.
Part 3: vertices
Tap each corner of a box with a fingertip. Each corner is a vertex, a point where edges meet. On a cube, three edges meet at every vertex: one along the top or bottom one way, one along the top or bottom the other way, and one upright.
The plan is the plainest of all. Every vertex of a cube is at the top or at the bottom: 4 on top and 4 on the bottom.
A small ball of playdough pressed onto each corner marks it. A square pyramid has 4 vertices around its base and 1 at the top point, 5 in all. Four edges meet at its top point, and three at each corner of its base.
Once a child can count all three parts, the counts become a way to name a solid without seeing it. A solid with 5 vertices, among these four, can only be the square pyramid. The table in the next section is the whole list.
Results
The counts for the four flat-faced solids.
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| cube | 6 | 12 | 8 |
| rectangular box | 6 | 12 | 8 |
| square pyramid | 5 | 8 | 5 |
| triangular prism | 5 | 9 | 6 |
The shapes of the faces. A cube has 6 squares, all the same size. A rectangular box has 6 rectangles, in matching pairs. A square pyramid has 1 square and 4 triangles. A triangular prism has 2 triangles and 3 rectangles.
Every edge belongs to exactly two faces. An edge is where two faces meet, so it has one face on each side, never more and never fewer.
A cube and a rectangular box have the same counts. Both have 6 faces, 12 edges and 8 vertices. What tells them apart is the shape of the faces: on a cube every face is a square of the same size.
The curved solids. In this lesson an edge is a straight line where two flat faces meet, and a vertex is a point where edges meet.
| Solid | Flat faces | Edges | Vertices |
|---|---|---|---|
| sphere | 0 | 0 | 0 |
| cylinder | 2 | 0 | 0 |
| cone | 1 | 0 | 0 |
All three have a curved surface. The flat faces of a cylinder and a cone are circles, and a cone also has a pointed tip, which is not a point where edges meet.
Why it is true
Top, bottom and around finds every face of a box once. A box sitting on a table has one face touching the table, the bottom, and one face facing straight up, the top. Every other face stands up from the table and goes around the box. So each face is in exactly one of the three groups, and counting the groups one after another counts each face once: 1 + 1 + 4 = 6.
Top, bottom and upright finds every edge of a cube once. An edge of a cube either lies flat along the top face, lies flat along the bottom face, or runs from the bottom up to the top. It cannot do two of these, since the top and the bottom never touch. The top square has 4 sides, the bottom square has 4 sides, and each of the 4 bottom corners has one edge going straight up from it. So there are 4 + 4 + 4 = 12 edges.
Counting the sides of every face counts each edge twice. Each face of a cube is a square with 4 sides, and every one of those sides is an edge of the cube. Adding the sides of all six faces gives
which is too many. The reason is the fold: every edge is a side of the two faces it joins, so every edge was counted once for each of them. Twelve edges, each counted twice, make 12 + 12 = 24. This is the cleanest way to see why a plan matters: a count with no plan is likely to land somewhere between 12 and 24.
Top and bottom finds every vertex of a cube once. Every corner of a cube is a corner of the top square or a corner of the bottom square, and not both. Each square has 4 corners, so there are 4 + 4 = 8 vertices.
The square pyramid. Every edge either goes around the square base or runs up to the top point, so there are 4 + 4 = 8. Every vertex is a corner of the base or the top point itself, so there are 4 + 1 = 5. Every face is the base or one of the triangles leaning in from a side of the base, one triangle for each of the 4 sides, so there are 1 + 4 = 5.
Worked examples
Part 1: painting the box fort
Oyatillo is building a box fort from big cardboard boxes, each taped shut. He wants to paint every face of one box a different color. How many colors does he need?
SolutionThe box is a rectangular box, and he needs one color for each face, so the question is how many faces it has. Count with the plan top, bottom, around. There is 1 face on top. Tip the box up and there is 1 face on the bottom. Now turn the box a quarter turn at a time and count the faces around the middle: front, one side, back, other side, which is 4. A sticky note on each face as it is counted shows when the count is finished, since the next turn brings back a face that already has a note.
Oyatillo needs 6 colors. The painted box also shows the matching pairs: the top and bottom are the same size, and so are the front and back, and the two sides.
Part 1: playdough prints
Mahliyo has a playdough triangular prism. She presses each face into a flat slab of playdough, once for each face. How many prints does she make, and what shapes are they?
Count by the ends and around. A triangular prism has two triangle ends, so 2 prints are triangles. Around the middle, between the two ends, the faces are rectangles: turn the prism and there are 3 of them before the first one comes back.
She makes 5 prints: 2 triangles and 3 rectangles. The two triangles match, because the ends are the same shape and size. A small dent pressed into each face after it is printed keeps her from printing one face twice.
Part 2: taping the edges
Oyatillo’s fort needs a strong box in the middle, a cube. He puts one strip of tape along each edge. How many strips does he need?
SolutionOne strip per edge, so count the edges, using the plan in the picture of the cube three times.
- Around the top face: 4 edges.
- Around the bottom face: 4 edges.
- Upright, from each bottom corner to the top: 4 edges.
He needs 12 strips. No edge is taped twice, because no edge is in two of the groups: an edge on top is not on the bottom, and an upright edge is not lying flat on either.
Part 2: snakes on a pyramid
Aperahama makes a square pyramid from playdough. He rolls thin playdough snakes and lays one along each edge. How many snakes does he need?
Count around the base first. The base is a square, and a square has 4 sides, so 4 edges go around the base. Then count the edges running up to the top point. There is one from each corner of the base, so 4 more.
He needs 8 snakes. A check: the pyramid has 1 square face and 4 triangle faces. Adding their sides gives 4 + 3 + 3 + 3 + 3 = 16, and since every edge is a side of two faces, that is each edge counted twice: 8 + 8 = 16.
Part 3: playdough balls on the corners
Mahliyo puts a small ball of playdough on every corner of her triangular prism. How many balls does she need?
Every corner of a triangular prism is a corner of one of its triangle ends. Each triangle has 3 corners, and the two ends share none.
She needs 6 balls. At each corner 3 edges meet: two around the triangle end and one long edge running across to the other end.
Part 3: solids with curved surfaces
Aperahama’s playdough set has a ball, a can shape and a cone. For each one, how many flat faces, edges and vertices does it have?
SolutionLook for flat surfaces first. The ball has none: every part of it is curved, and it rolls every way. The can has 2 flat faces, the circles on its top and bottom. The cone has 1 flat face, the circle it stands on.
Now the edges. An edge is a straight line where two flat faces meet. On the can, the two circles never touch each other; each one meets only the curved side, along a round rim. On the cone, the circle meets only the curved side. So none of the three has an edge in this lesson’s count, and with no edges there is no point where edges meet, so none has a vertex.
| Solid | Flat faces | Edges | Vertices |
|---|---|---|---|
| ball | 0 | 0 | 0 |
| can | 2 | 0 | 0 |
| cone | 1 | 0 | 0 |
The cone still has a pointed tip, and it is fine to call it that. It is a point, but not a point where edges meet. The rims and the tip are taken up again under Boundaries.
Part 3: guess the solid
Oyatillo hides a solid from his fort behind his back and gives clues. His first clue: “It has 5 vertices, and one of its faces is a square.” Which solid is it? Then he hides another: “It has 6 faces, 12 edges and 8 vertices, and not every face is a square.” Which solid is that?
SolutionUse the table of counts. For the first solid, look down the vertices column: 8, 8, 5, 6. Only the square pyramid has 5. Check the other clue too: its base is a square, so it fits.
For the second solid, 6 faces, 12 edges and 8 vertices fit both the cube and the rectangular box. The counts alone cannot decide. The last clue does: every face of a cube is a square, and this solid has a face that is not. So it is a rectangular box. A clue about the shape of a face settled what the counts could not.
Boundaries
- Faces must be flat. A curved surface, like the side of a can, is not a face here. Some books call it a “curved face” or a “curved surface”; either way it is not counted with the flat faces, and this lesson’s counts are for flat faces only.
- Edges here are straight and join two flat faces. The round rim of a can is where a flat circle meets the curved side. Some books call each rim a “curved edge” and say a cylinder has 2 edges. Both ways of counting are in use, so when a count matters, say which one: in this lesson, a can has 0 edges.
- A cone’s tip is a point, not a corner where edges meet. Many books call it the cone’s vertex, since it is a pointed tip, and later work with cones uses that word. No edges meet there, so it is not counted with the corners of a box.
- The counts belong to a whole solid, not to what can be seen. A drawing or a box on a table hides some faces, edges and corners. The dashed lines in the pictures stand for hidden edges. Counting only what can be seen from one place gives too few.
- Gluing solids together changes the counts. Two cubes pressed face to face make a tall rectangular box with 6 faces, 12 edges and 8 vertices, not twice the cube’s counts. The two faces pressed together are now inside, and the squares that line up side by side become one flat face (Exercise 25).
- The counts do not fix the solid. A cube and a rectangular box share all three counts. A face shape, a length or a picture is needed to tell them apart.
Where this goes next
Faces lead straight back to flat shapes. Each face is a square, rectangle or triangle, and drawing shapes from their attributes uses the same kind of counting, of sides and corners, on the flat page.
Later, the four solids here grow into families. Prisms and pyramids come with any shape of base, and their counts follow a pattern from the base: a prism whose ends have 5 sides, or a pyramid on a hexagon, can be counted without ever being built. Solids with only flat faces meet a famous fact about these three counts, Euler’s formula: for every solid of this kind without a hole through it, the vertices take away the edges, plus the faces, always comes to 2. An adult can check it on the cube now: 8 - 12 + 6 = 2. The five regular solids are the ones whose faces are all the same regular shape with the same number meeting at every vertex. The curved solids have lessons of their own, cylinders and cones and Spheres.
Faces are also where surface measurements start. Unfold a box along some of its edges and its faces lie flat in one piece, a net (Nets of solids), and adding up the faces gives Surface area.
Exercises
Part 1: faces
- How many faces does a cube have? What shape is every face?
- How many faces does a square pyramid have? What shapes are they?
- A number cube has a different number of dots on each face, starting with 1 dot and going up by one each time. What is the largest number of dots on any face?
- Aperahama presses every flat face of a playdough can into a slab. How many prints does he make, and what shape are they?
- Oyatillo is covering each face of a rectangular box with one sheet of paper. He has covered the top and the front. How many more sheets does he need?
- How many faces does solid A in the picture have? What shape are they?
Part 2: edges
- How many edges does a rectangular box have? Count around the top, around the bottom and upright.
- How many edges does a triangular prism have?
- How many edges does solid A have?
- Oyatillo has put tape along 5 edges of a cube box. How many more strips of tape does he need to tape every edge?
- Which has more edges, a cube or a triangular prism? How many more?
Part 3: vertices
- How many vertices does a cube have?
- How many vertices does a square pyramid have?
- Mahliyo is putting a playdough ball on every corner of a rectangular box. She has put on 3. How many more does she need?
- How many vertices does solid A have?
- A cone has how many flat faces? What shape is it? How many edges does a cone have?
- Which of these have no vertices: a cube, a ball, a can, a square pyramid?
Part 3: name the solid
- A solid has 6 faces, and every face is a square of the same size. Name it.
- Among the four flat-faced solids of this lesson, which one has 9 edges?
- Among the four flat-faced solids of this lesson, which one has 8 edges?
- Solid B in the picture is shaped like a house. How many faces, edges and vertices does it have?
Explain
- A cube and a rectangular box both have 6 faces, 12 edges and 8 vertices. How can you tell them apart?
- Add up the sides of every face of a cube: 4 + 4 + 4 + 4 + 4 + 4. Why is the total twice the number of edges?
- A can has 2 flat faces. Why does it have no edges in this lesson’s count?
- Oyatillo stacks two cube boxes of the same size, one on the other, and tapes them together so they make one tall box. How many faces, edges and vertices does the tall box have? Why is it not twice the cube’s counts?
Answers
- 6 faces, and every face is a square, all of the same size. Top, bottom, around: 1 + 1 + 4 = 6.
- 5 faces: 1 square, the base, and 4 triangles, one leaning in from each side of the base: 1 + 4 = 5.
- 6 dots. A number cube has 6 faces, one number for each, so the numbers go 1 to 6.
- 2 prints, both circles: one from the top and one from the bottom. The side is curved, so it is not a face and leaves no flat print.
- 4 more sheets. A rectangular box has 6 faces and 2 are covered, and 2 + 4 = 6.
- 4 faces, all triangles: 1 on the bottom and 3 leaning in to the top point, 1 + 3 = 4.
- 12 edges: 4 around the top, 4 around the bottom and 4 upright, 4 + 4 + 4 = 12.
- 9 edges: 3 around one triangle end, 3 around the other, and 3 long edges joining the ends, 3 + 3 + 3 = 9.
- 6 edges: 3 around the triangle base and 3 running up to the top point, 3 + 3 = 6.
- 7 more strips. A cube has 12 edges and 5 are taped, and 5 + 7 = 12.
- The cube, by 3. A cube has 12 edges and a triangular prism has 9, and 9 + 3 = 12.
- 8 vertices: 4 around the top and 4 around the bottom, 4 + 4 = 8.
- 5 vertices: 4 at the corners of the base and 1 at the top point, 4 + 1 = 5.
- 5 more. A rectangular box has 8 corners and 3 have a ball, and 3 + 5 = 8.
- 4 vertices: 3 at the corners of the base and 1 at the top point, 3 + 1 = 4.
- 1 flat face, a circle. A cone has 0 edges: its circle meets only the curved side, never another flat face. Its tip is a point, but no edges meet there.
- The ball and the can. A cube has 8 vertices and a square pyramid has 5. The ball has no edges at all, and the can’s two flat faces never meet each other, so neither has a point where edges meet.
- A cube.
- The triangular prism. The others have 12, 12 and 8 edges.
- The square pyramid. The others have 12, 12 and 9 edges.
- 7 faces, 15 edges and 10 vertices. Faces: the bottom, 2 long walls, 2 roof slopes, and the front and the back, each one five-sided face, 1 + 2 + 2 + 2 = 7. Edges: 4 around the bottom, 4 upright, 2 where the walls meet the roof, 4 sloping up to the ridge, and the ridge itself, 4 + 4 + 2 + 4 + 1 = 15. Vertices: 4 on the bottom, 4 at the tops of the walls and 2 at the ends of the ridge, 4 + 4 + 2 = 10.
- Look at the shapes of the faces. Every face of a cube is a square, all the same size. A rectangular box that is not a cube has at least one face that is not a square.
- The total is 24, and a cube has 12 edges. Every edge is where two faces meet, so it is a side of both of them and was added once for each: 12 + 12 = 24.
- An edge is a straight line where two flat faces meet. The can’s two circles never touch each other; each one meets only the curved side, along a round rim. So no two flat faces meet, and there is no edge. Some books call the rims curved edges, and then a can has 2.
- 6 faces, 12 edges and 8 vertices, the same as any rectangular box. The two faces pressed together are inside the tower now, so they are not faces of it. On each side, the square of the top cube and the square of the bottom cube line up flat and make one tall rectangle face, so the 4 edges around the middle are no longer edges, and the 4 corners around the middle are no longer corners. Top, bottom, around: 1 + 1 + 4 = 6 faces; 4 + 4 + 4 = 12 edges; 4 + 4 = 8 vertices.