2D shapes: flat figures
A 2D shape is flat. It has length and width, but no depth.
A polygon is a closed flat shape made of straight sides. We name it by its number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), octagon (8).
A regular polygon has all sides equal and all angles equal. A square is a regular quadrilateral.
The angles inside a polygon with n sides add up to (n − 2) × 180°. A triangle: 180°. A quadrilateral: 360°. A hexagon: 720°, so each angle of a regular hexagon is 720° ÷ 6 = 120°.
A circle is not a polygon. It has a centre, a radius and a curved edge.
3D solids: faces, edges and vertices
A 3D shape (solid) has length, width and depth.
- Face: a flat surface.
- Edge: a line where two faces meet.
- Vertex (plural vertices): a corner where edges meet.
A prism has the same shape all the way through, with two equal ends. A pyramid has one base and triangle faces that meet at a top point. Cylinders, cones and spheres have curved surfaces.
For any solid with flat faces and no holes, Euler's rule works: F + V − E = 2. Cube: 6 + 8 − 12 = 2.
Showing a solid on flat paper
We can draw a solid as a net (unfolded pattern), as views (front, side, top), or as a 3D sketch on dotted (isometric) paper. A cube has 11 different nets.
Congruent and similar shapes
Congruent shapes are exact copies: same shape, same size. You can slide, turn or flip one to sit exactly on the other.
Similar shapes have the same shape but may be a different size. All angles are equal and all lengths are multiplied by the same scale factor k.
The famous Pythagoras rule connects the sides of a right triangle: a² + b² = c². A 3-4-5 triangle and a 6-8-10 triangle are similar (k = 2), and both obey it.
Tessellations and transformations
A tessellation is a pattern of shapes that covers a surface with no gaps and no overlaps.
It works when the angles meeting at each point add up to exactly 360°. Squares: 4 × 90°. Equilateral triangles: 6 × 60°. Regular hexagons: 3 × 120°. A regular pentagon (108°) cannot tile alone, because 360 ÷ 108 is not a whole number.
Tiles are moved by transformations: translation (slide), rotation (turn) and reflection (flip). These moves keep shapes congruent.
Scale drawings and models
A scale such as 1 : 100 means 1 cm on paper stands for 100 cm (1 m) in real life.
Real length = drawing length × 100. Drawing length = real length ÷ 100.
When every length is multiplied by k: area is multiplied by k² and volume by k³. Double a cube's side and it needs 4 times the paint and holds 8 times the water.
Try it at home
Draw the cross-shaped net of a cube on card (six 5 cm squares). Cut, fold and tape it. Count the faces, edges and vertices. Then check F + V − E. Next, look at your floor or a footpath: which shapes tessellate? Measure one tile and work out how many fill a 2 m × 2 m area.
Key formulas and definitions
- Angle sum of a polygon with n sides = (n − 2) × 180°
- Euler's rule for solids: F + V − E = 2
- Tessellation: angles at each point add to 360°
- Pythagoras: a² + b² = c² (right triangle, c is the longest side)
- Scale 1 : n → real length = drawing length × n
- Scale factor k → length ×k, area ×k², volume ×k³
Worked examples
1. How many faces, edges and vertices does a triangular prism have? Check Euler's rule.
Faces: 2 triangles + 3 rectangles = 5. Edges: 3 + 3 + 3 = 9. Vertices: 3 + 3 = 6. Check: F + V − E = 5 + 6 − 9 = 2 ✓
2. Find each angle of a regular octagon.
Sum = (8 − 2) × 180° = 1080°. Each angle = 1080° ÷ 8 = 135°.
3. Can regular octagons tessellate on their own?
Each angle is 135°. 360 ÷ 135 = 2.67, not a whole number. So no. (Octagons plus small squares do work: 135 + 135 + 90 = 360.)
4. On a 1 : 50 plan a room is 8 cm by 6 cm. What is the real size and floor area?
8 × 50 = 400 cm = 4 m. 6 × 50 = 300 cm = 3 m. Area = 4 × 3 = 12 m².
5. Triangle A has sides 5, 12, 13 cm. Triangle B is similar with shortest side 15 cm. Find the other sides.
k = 15 ÷ 5 = 3. Sides: 12 × 3 = 36 cm, 13 × 3 = 39 cm.
6. A model car is built with scale factor k = 1/20. The real car's roof area is 4 m². What is the model's roof area?
Area scales by k² = 1/400. 4 m² ÷ 400 = 0.01 m² = 100 cm².
Common mistakes
- Mixing up edges and vertices. Edges are lines; vertices are the corner points.
- Thinking any 6 squares joined together make a cube net. Some patterns overlap when folded; test by folding.
- Calling two shapes similar just because they look alike. All angles must match and all sides must have the same ratio.
- Multiplying area by k instead of k² when scaling. A 2× bigger square has 4× the area.