Geometry and Shapes — 2D and 3D
1. Basic Geometric Terms
Point:
A POINT is an exact location. It has NO size — only position. 'Mark it with a dot and name it with a capital letter: ● A'
Line:
A LINE is a straight path of points that goes on FOREVER in both directions. 'Lines have NO endpoints. We show them with arrows on both ends: ⟷'
Line Segment:
A LINE SEGMENT is a part of a line with TWO endpoints. 'We can MEASURE a line segment. It has a definite length: ●——●'
Ray:
A RAY has ONE endpoint and goes on FOREVER in ONE direction. 'A ray starts at a point and goes in one direction: ●——⟶'
| Term | Endpoints | Length | Picture |
|---|---|---|---|
| Point | None | No size | ● |
| Line | None (infinite) | Unlimited | ⟷ |
| Line Segment | Two | Fixed | ●—● |
| Ray | One | Unlimited in one direction | ●—⟶ |
2. Angles
An ANGLE is formed when TWO rays meet at a common endpoint (called the VERTEX).
Types of Angles:
| Type | Size | Looks like | Example |
|---|---|---|---|
| Right angle | Exactly 90° | Corner of a book | L |
| Acute angle | Less than 90° | Small, sharp corner | < |
| Obtuse angle | More than 90° | Wide, open corner | > (wide) |
| Straight angle | Exactly 180° | A straight line | — |
'Check for a RIGHT angle using a set square or the corner of a notebook. If it is SMALLER, it is ACUTE. If it is BIGGER, it is OBTUSE.'
3. 2D Shapes (Plane Shapes)
TWO-DIMENSIONAL shapes have LENGTH and WIDTH but NO thickness or depth.
Circle:
- A closed curve with EVERY point at the SAME distance from the CENTRE.
- Radius (r): Distance from centre to any point on the circle.
- Diameter (d): Distance across the circle through the centre. d = 2 × r
Triangle:
- A shape with 3 SIDES and 3 ANGLES.
- Sum of interior angles = 180°.
- Types: Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides).
Square:
- 4 EQUAL sides.
- 4 RIGHT angles (90° each).
- Opposite sides are PARALLEL.
Rectangle:
- 4 sides — OPPOSITE sides equal.
- 4 RIGHT angles (90° each).
- Opposite sides are PARALLEL.
Polygon:
A POLYGON is a closed shape made of STRAIGHT line segments.
| Shape | Number of Sides | Number of Angles | Name |
|---|---|---|---|
| Triangle | 3 | 3 | Trigon |
| Quadrilateral | 4 | 4 | Tetragon |
| Pentagon | 5 | 5 | Pentagon |
| Hexagon | 6 | 6 | Hexagon |
| Octagon | 8 | 8 | Octagon |
2D Shape Properties Table:
| Shape | Sides | Corners | Lines of Symmetry |
|---|---|---|---|
| Circle | 0 (curved) | 0 | Infinite |
| Triangle | 3 straight | 3 | Depends on type |
| Square | 4 equal straight | 4 (90°) | 4 |
| Rectangle | 4 (opposite equal) | 4 (90°) | 2 |
| Pentagon | 5 straight | 5 | 5 |
| Hexagon | 6 straight | 6 | 6 |
4. 3D Shapes (Solid Shapes)
THREE-DIMENSIONAL shapes have LENGTH, WIDTH, and DEPTH (or height).
Key Terms:
- Face: A flat surface of a 3D shape.
- Edge: Where TWO faces meet.
- Vertex (vertices): Where THREE or more edges meet (corners).
Cube:
- 6 square faces (ALL equal).
- 12 edges (all equal length).
- 8 vertices.
- 'Think of a DICE or a RUBIK'S CUBE.'
Cuboid:
- 6 rectangular faces (opposite faces equal).
- 12 edges.
- 8 vertices.
- 'Think of a BRICK or a SHOE BOX.'
Sphere:
- 1 curved surface (NO flat face).
- NO edges and NO vertices.
- 'Think of a BALL or a GLOBE.'
Cylinder:
- 2 flat circular faces (top and bottom).
- 1 curved surface.
- 2 edges (where the flat faces meet the curved surface).
- NO vertices.
- 'Think of a TIN CAN or a COLD DRINK CAN.'
Cone:
- 1 flat circular face (base).
- 1 curved surface.
- 1 curved edge.
- 1 vertex (the tip).
- 'Think of an ICE CREAM CONE or a PARTY HAT.'
5. Faces, Edges, and Vertices Table
| Solid Shape | Faces | Edges | Vertices | Shape of Faces |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | 6 squares |
| Cuboid | 6 | 12 | 8 | 6 rectangles |
| Sphere | 1 (curved) | 0 | 0 | — |
| Cylinder | 3 (2 flat + 1 curved) | 2 | 0 | 2 circles |
| Cone | 2 (1 flat + 1 curved) | 1 | 1 | 1 circle |
| Triangular Prism | 5 | 9 | 6 | 2 triangles + 3 rectangles |
| Square Pyramid | 5 | 8 | 5 | 1 square + 4 triangles |
6. Nets of 3D Shapes
A NET is a FLAT pattern that can be FOLDED to make a 3D shape.
'If you cut open a cardboard box and lay it flat — THAT is its net!'
| Shape | Net Description |
|---|---|
| Cube | 6 connected squares in a cross shape |
| Cuboid | 6 connected rectangles |
| Cylinder | 2 circles + 1 rectangle |
| Cone | 1 circle + 1 sector (like a pizza slice) |
7. Common Mistakes
- Confusing 2D and 3D: 'A square is FLAT (2D). A cube is SOLID (3D). A square has area. A cube has volume.'
- Counting faces on a cylinder: 'A cylinder has 3 faces — 2 flat circles and 1 curved surface. Many students forget the curved surface!'
- Circle is NOT a polygon: 'A polygon must have STRAIGHT sides. A circle has a curved boundary — so it is NOT a polygon.'
- Vertex vs. corner: 'Vertex is a corner where edges meet — in both 2D and 3D shapes. A cube has 8 vertices.'
8. Key Facts to Remember
- 'Every polygon is a closed shape made of straight line segments.'
- 'A square is a SPECIAL type of rectangle (with all sides equal).'
- 'The sum of angles in a triangle is ALWAYS 180°.'
- 'A sphere has NO edges, NO vertices, and 1 curved face.'
- 'A cube is a special type of cuboid where all sides are EQUAL.'
9. Self-Test
Q1: Name a 2D shape with: (a) 3 sides (b) 4 equal sides and 4 right angles (c) 5 sides
Q2: Name a 3D shape that has 6 faces, 12 edges, and 8 vertices. Is it a cube or a cuboid? How can you tell?
Q3: How many faces does a sphere have?
Q4: What is the difference between a line and a ray?
Q5: Draw a net for a cube (describe it in words).
Q6: Which 3D shape has 1 vertex, 1 edge, 1 circular face, and 1 curved surface?
Q7: True or False: A circle is a polygon.
Q8: Name an object around you that is (a) cuboid-shaped (b) cylinder-shaped.
Answers:
A1: (a) Triangle (b) Square (c) Pentagon A2: Both have 6 faces, 12 edges, and 8 vertices. If all faces are equal squares → Cube. If faces are rectangles → Cuboid. A3: 1 curved face (no flat faces). A4: A line goes on forever in BOTH directions. A ray goes on forever in ONE direction from an endpoint. A5: A cross shape made of 6 connected squares — 4 in a row and 1 above the second and 1 below the second. A6: Cone. A7: False — a circle has a curved boundary, not straight sides. A8: (a) A book (cuboid) (b) A cold drink can (cylinder)
