What you'll learn
A plane shape is flat and has only length and width — a square, a triangle, a circle drawn on paper. A solid takes up space and has length, width and height — a box, a ball, a tin. Plane shapes are described by their sides and corners; solids are described by their faces (the flat or curved surfaces), their edges (where two faces meet) and their vertices (the corners where edges meet). This topic covers naming plane shapes by their number of sides, sorting triangles and quadrilaterals, naming the common solids, counting faces, edges and vertices, and recognising a solid from its net. Geometry carries 6 of the 40 items on the SEA Mathematics paper, and "how many edges has this solid" is one of the most reliable marks on the whole paper — because the answer is a fact you can simply learn.
Key terms and definitions
Plane shape — a flat, two-dimensional shape such as a square or a circle.
Solid — a three-dimensional object such as a cube, a cylinder or a sphere.
Face — a surface of a solid. It may be flat, like the side of a box, or curved, like the side of a tin.
Edge — the straight or curved line where two faces meet.
Vertex — a corner point where edges meet. The plural is vertices.
Polygon — a closed plane shape with straight sides only. A circle is not a polygon.
Regular polygon — a polygon whose sides and angles are all equal, such as a square or an equilateral triangle.
Prism — a solid with two identical ends and flat rectangular sides joining them.
Pyramid — a solid with one flat base and triangular faces that meet at a single point at the top.
Net — a flat pattern that can be folded up to make a solid.
Radius and diameter — the distance from the centre of a circle to its edge, and the distance right across through the centre. The diameter is twice the radius.
Core concepts
Naming plane shapes by their sides
A polygon is named by counting its sides.
| Sides | Name |
|---|---|
| 3 | triangle |
| 4 | quadrilateral |
| 5 | pentagon |
| 6 | hexagon |
| 7 | heptagon |
| 8 | octagon |
Every polygon has the same number of corners as it has sides. A hexagon has 6 sides and 6 corners.
The four kinds of triangle
| Triangle | What makes it that |
|---|---|
| Equilateral | all three sides equal, all three angles equal |
| Isosceles | exactly two sides equal, and the two angles opposite them equal |
| Scalene | all three sides different |
| Right-angled | one angle is a square corner of 90° |
A triangle can be right-angled and isosceles at the same time, so do not assume the four names never overlap.
Sorting the quadrilaterals
All six of these have four sides, and telling them apart is a favourite exam item.
| Quadrilateral | Description |
|---|---|
| Square | four equal sides, four right angles |
| Rectangle | opposite sides equal, four right angles |
| Rhombus | four equal sides, opposite angles equal, but no right angles needed |
| Parallelogram | opposite sides equal and parallel, opposite angles equal |
| Trapezium | exactly one pair of parallel sides |
| Kite | two pairs of equal sides that are next to each other, not opposite |
A square is a special rectangle, one where all the sides became equal, and it is also a special rhombus, one where the angles became right angles.
The circle
A circle is a plane shape with no straight sides and no corners. The centre is the middle point; the radius runs from the centre to the edge; the diameter runs right across through the centre and is 2 × radius; the circumference is the distance all the way around the edge. If a plate has a radius of 12 cm, its diameter is 24 cm.
The common solids, and their faces, edges and vertices
These are the numbers to learn. There is no working out to do; the marks come from knowing them.
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
| Triangular pyramid | 4 | 6 | 4 |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
A cube and a cuboid have the same counts because a cube is simply a cuboid whose edges are all equal.
The cylinder, cone and sphere are the odd ones out because they have curved surfaces. A tin of milk has a circle at the top, a circle at the bottom and one curved surface wrapped round; its only edges are the two circles at the ends, and it has no corners at all.
A check that always works
For any solid with flat faces, faces plus vertices always equals edges plus two.
Test it on the triangular prism: 5 faces + 6 vertices = 11, and 9 edges + 2 = 11. On the square-based pyramid: 5 + 5 = 10, and 8 + 2 = 10. Both match.
If you have counted a solid and the two sides do not agree, you have miscounted something.
Nets
A net is what a solid looks like unfolded flat. You can recognise the solid by counting the shapes in the net.
| Net made of | Folds up into |
|---|---|
| 6 squares | a cube |
| 6 rectangles | a cuboid |
| 2 triangles and 3 rectangles | a triangular prism |
| 1 square and 4 triangles | a square-based pyramid |
| 2 circles and 1 rectangle | a cylinder |
The number of shapes in a net is always the number of faces.
Worked examples
Example 1: Counting a triangular prism
Anisa has a tent shaped like a triangular prism. How many faces, edges and vertices does it have?
Take the parts in turn. The two ends are triangles: that is 2 faces. The floor and the two sloping sides are rectangles: that is 3 more faces. Total 5 faces.
Each triangle has 3 edges, giving 3 + 3 = 6, and there are 3 more edges running the length of the tent joining the two triangles. Total 9 edges.
Each triangle has 3 corners, and there are two of them. Total 6 vertices.
Check with the rule: 5 faces + 6 vertices = 11, and 9 edges + 2 = 11. They agree, so the counts are right.
Example 2: Naming a solid from a description
Kern is told: "This solid has 5 faces. One face is a square and the other four are triangles that meet at a point." Name the solid and say how many edges and vertices it has.
Triangles meeting at a single point means it is a pyramid, and the flat face it stands on is a square, so it is a square-based pyramid.
The edges: the square base has 4 edges, and 4 more edges run up from the corners of the square to the point at the top. That is 4 + 4 = 8 edges.
The vertices: the 4 corners of the square, plus the point at the top. That is 5 vertices.
Check: 5 + 5 = 10, and 8 + 2 = 10. Correct.
A triangular pyramid, whose base is a triangle instead, has 4 faces, 6 edges and 4 vertices. The word before "pyramid" tells you the base, and the base tells you everything else.
Example 3: Matching solids to objects
Riya empties a shopping bag onto the table. Name the solid that each item is most like, and give the number of vertices for each.
| Item | Solid | Vertices |
|---|---|---|
| A tin of corned beef | cylinder | 0 |
| A dice from a board game | cube | 8 |
| A box of cornflakes | cuboid | 8 |
| An orange | sphere | 0 |
| A paper cup for snow cone | cone | 1 |
The tin and the orange catch pupils out. It feels wrong that a solid can have no corners at all, but a vertex is where edges meet, and a sphere has no edges. The cone has exactly one vertex, at its point, and one edge, the circle round its open end.
Common mistakes and how to avoid them
Mixing up edges and faces. Answering "6" for the edges of a cube. Fix: a face is a surface you could paint; an edge is a line you could run your finger along. A cube has 6 faces and 12 edges.
Counting the edges of a cube from one side only. Fix: count the 4 edges around the top, the 4 around the bottom and the 4 upright ones. 4 + 4 + 4 = 12.
Saying a cylinder has 2 faces. Fix: the curved surface counts too, so a cylinder has 3.
Giving a sphere edges or vertices. Fix: it has one curved surface and nothing else.
Thinking a square is not a rectangle. Fix: a square meets every condition of a rectangle, so it is a special rectangle. The exam may accept either name, but "square" is the more exact answer.
Confusing a rhombus with a parallelogram. Fix: a rhombus needs all four sides equal; a parallelogram only needs opposite sides equal.
Confusing a pyramid with a prism. Fix: a prism stays the same width all along; a pyramid tapers to a point.
Calling a solid by a plane-shape name. Answering "square" for a dice or "circle" for a ball. Fix: flat shapes are squares and circles; solids are cubes and spheres.
How parents can help at home
Nothing here needs pen and paper. It needs a cupboard.
Do a kitchen sort. Take out six items — a tin, a cereal box, a ball, a funnel, a dice, a tub — and ask your child to name each solid, then to give the faces, edges and vertices of the ones with flat sides.
Count by touching. Ask your child to run a finger along each edge as they count, tallying out loud. Miscounting almost always comes from counting the same edge twice, and touching stops it.
Unfold a box. Open out an empty cereal box or a juice carton flat. That is a net, and seeing it happen once teaches more than any explanation. Ask how many rectangles there are, and check it against the 6 faces of a cuboid.
Use the right word. Say "cube" rather than "square", and "sphere" rather than "circle", when talking about an object. Children lose marks for this more often than for anything else in Geometry.
When your child is stuck, the prompt is "say out loud what the ends look like and what the sides look like". Two triangles and three rectangles names itself. If they are stuck counting, ask them to count the top, then the bottom, then the uprights, as three separate numbers.
To check an answer without doing any geometry yourself, use the rule: faces plus vertices should equal edges plus two. If it does not, ask them to count again.
Exam technique for solids and plane shapes
Learn the table of faces, edges and vertices by heart. These items take ten seconds each if you know the numbers and two anxious minutes if you do not.
Read whether the item is about a flat shape or a solid, and answer with a matching name.
When counting edges, count them in groups — the top set, the bottom set, the upright set — and write the three numbers before adding.
Use the faces + vertices = edges + 2 check whenever you have time. It catches nearly every counting slip.
For a net item, count how many of each shape there are and match it to the table. Six squares can only be a cube.
Watch for the word "curved". A curved surface still counts as a face, and questions often ask for "flat faces" only, which is a different number.
Write the plural properly: one vertex, many vertices.
Quick revision summary
- Plane shapes are flat; solids take up space. Polygons are named by their sides: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 8 octagon.
- Triangles: equilateral (3 equal sides), isosceles (2 equal), scalene (none equal), right-angled (one 90° angle).
- A square is a special rectangle and a special rhombus.
- A trapezium has exactly one pair of parallel sides; a kite has two pairs of equal neighbouring sides.
- Cube and cuboid: 6 faces, 12 edges, 8 vertices.
- Triangular prism: 5 faces, 9 edges, 6 vertices. Square-based pyramid: 5 faces, 8 edges, 5 vertices. Triangular pyramid: 4 faces, 6 edges, 4 vertices.
- Cylinder: 3 faces, 2 edges, 0 vertices. Cone: 2 faces, 1 edge, 1 vertex. Sphere: 1 face, 0 edges, 0 vertices.
- Faces + vertices = edges + 2 for any solid with flat faces.
- A net has one flat shape for each face: 6 squares make a cube; 2 circles and a rectangle make a cylinder.