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In Key Stage 2 you go much deeper into the properties of 2D and 3D shapes. You learn about angles in triangles and quadrilaterals, how to classify shapes precisely, and how to calculate angles you haven't been given.
All triangles have 3 sides and 3 angles, and the angles always add up to 180 degrees.
| Type | Properties |
|---|---|
| Equilateral | 3 equal sides, 3 equal angles (each 60 degrees) |
| Isosceles | 2 equal sides, 2 equal base angles |
| Scalene | All sides different, all angles different |
| Right-angled | One angle = 90 degrees (the other two add up to 90) |
Finding a missing angle in a triangle:
All quadrilaterals have 4 sides, and their interior angles always add up to 360 degrees.
| Shape | Properties |
|---|---|
| Square | 4 equal sides, 4 right angles |
| Rectangle | 2 pairs of equal sides, 4 right angles |
| Parallelogram | 2 pairs of equal and parallel sides, opposite angles equal |
| Rhombus | 4 equal sides, opposite sides parallel, opposite angles equal |
| Trapezium | Exactly 1 pair of parallel sides |
| Kite | 2 pairs of adjacent equal sides, 1 pair of equal angles |
Finding a missing angle in a quadrilateral:
Use this tree to identify a quadrilateral from its properties:
flowchart TD
A["Quadrilateral<br/>4 sides, angles sum 360"] --> B{All 4 sides equal?}
B -->|Yes| C{All angles 90?}
B -->|No| D{"2 pairs of equal<br/>parallel sides?"}
C -->|Yes| E[Square]
C -->|No| F[Rhombus]
D -->|Yes| G{All angles 90?}
D -->|No| H{"Exactly 1 pair<br/>of parallel sides?"}
G -->|Yes| I[Rectangle]
G -->|No| J[Parallelogram]
H -->|Yes| K[Trapezium]
H -->|No| L["Kite or<br/>irregular quadrilateral"]
A polygon is any flat shape with straight sides.
A regular polygon has all sides equal and all angles equal.
Sum of interior angles of any polygon:
Sum = (number of sides - 2) x 180 degrees
A circle is a perfectly round 2D shape. Key vocabulary:
| Term | Meaning |
|---|---|
| Centre | The middle point of the circle |
| Radius | The distance from the centre to the edge (r) |
| Diameter | The distance across the circle through the centre (d = 2r) |
| Circumference | The distance all the way around the outside of the circle |
Relationship: diameter = 2 x radius. So if radius = 5 cm, diameter = 10 cm.
The circumference is approximately 3.14159... times the diameter. This number is called pi (written as pi, approximately 3.14).
A net is a flat shape that folds up to make a 3D shape.
Cube net: 6 squares arranged in a cross or T-shape (there are 11 different valid nets for a cube).
Cuboid net: 6 rectangles arranged so that opposite faces are identical.
Key 3D shape properties:
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Cylinder | 3 | 2 | 0 |
| Cone | 2 | 1 | 1 |
| Square-based pyramid | 5 | 8 | 5 |
| Triangular prism | 5 | 9 | 6 |
| Sphere | 1 | 0 | 0 |
Euler's relationship: For any polyhedron (a 3D shape with flat faces):
A shape has a line of symmetry (also called a mirror line) if you can fold it so that both halves match exactly.
Number of lines of symmetry for common shapes:
| Shape | Lines of symmetry |
|---|---|
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Square | 4 |
| Rectangle | 2 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinite |
| Parallelogram | 0 |
| Kite | 1 |
Key rule: A regular polygon with n sides has n lines of symmetry.
Completing a symmetric figure: If you are given half a shape and a mirror line, reflect each point to the other side:
Example: A point is 3 squares to the left of a vertical mirror line. Its reflection is 3 squares to the right of the mirror line.
Parallel lines are lines that never meet — they are always the same distance apart. Marked with arrows on diagrams.
Perpendicular lines meet at a right angle (90 degrees). Marked with a small square at the corner.
A SATs-style question shows a triangle ABC sitting on a straight line. Angle BAC at vertex A is 70 degrees, angle ABC at vertex B is 50 degrees, and the line continues past C to a point D. Find angle BCD (the exterior angle at C).
Step 1 — Find the unknown interior angle at C. The interior angles of a triangle add to 180 degrees.
angle BCA = 180 - 70 - 50 = **60 degrees**
Step 2 — Use the straight-line rule. Angles on a straight line sum to 180 degrees. The straight line BCD passes through C, so the interior angle BCA and the exterior angle BCD are supplementary.
angle BCD = 180 - 60 = **120 degrees**
Step 3 — Check using the exterior-angle property. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
angle BCD = 70 + 50 = **120 degrees** ✓
Both methods agree, which confirms the answer.
Step 4 — Extend the problem with a quadrilateral. Suppose the diagram now adds a fourth vertex E to make quadrilateral ABCE, where angle BAE = 70, angle ABC = 50, angle BCE = 60 (the interior angle at C in the quadrilateral, which equals 60 by the diagram), and we want the angle at E.
Sum of angles in a quadrilateral = 360 degrees.
angle AEC = 360 - 70 - 50 - 60 = **180 degrees**
A 180-degree interior angle is a straight angle — meaning E is not really a vertex; the line is straight at that point. So in fact ABCE is not a true quadrilateral; it would simplify to a triangle. This kind of reasoning shows pupils that angle calculations have meaning, not just numerical totals.
Step 5 — Apply to a regular polygon. What is each interior angle of a regular hexagon?
Sum of interior angles = (6 - 2) x 180 = 720 degrees
Each interior angle = 720 / 6 = **120 degrees**
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