Geometric language, 2D and 3D shapes

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Notes de leçon

Key Geometric Terms and Notation

Point: an exact location, named with a capital letter (e.g., A).
Line: a straight path extending infinitely both ways; line segment has two endpoints; ray has one endpoint.
Vertex (plural vertices): a corner where two or more edges meet; edge: a line segment where two faces meet.
Plane: a flat surface that extends infinitely; a face of a 3D shape lies in a plane.
Parallel lines are distinct coplanar lines that never meet. Perpendicular lines meet at 90°, marked with a small square.
Congruent shapes have identical size and shape. Similar shapes have corresponding lengths in a constant ratio; scale factor 1 is allowed.
Regular polygon: all sides equal and all interior angles equal.

Similar triangles

Properties of 2D Shapes

Triangle types: equilateral (3 equal sides, 3 equal angles), isosceles (2 equal sides, 2 equal angles), scalene (no equal sides), right-angled (one 90° angle).
Quadrilateral types: square (4 equal sides, 4 right angles), rectangle (opposite sides equal, 4 right angles), parallelogram (opposite sides parallel and equal), rhombus (4 equal sides, opposite angles equal), trapezium (one pair of parallel sides), kite (two pairs of adjacent equal sides).
Polygon naming: 3 sides = triangle, 4 = quadrilateral, 5 = pentagon, 6 = hexagon, 8 = octagon.
A regular polygon has all sides and angles equal; e.g., regular hexagon has 6 equal sides and 6 equal angles.
Interior angle sum of an n-sided polygon = (n − 2) × 180°.
Diagonals: lines connecting non-adjacent vertices; a square has 2 diagonals, a pentagon has 5.

Regular hexagon

Faces, Edges, and Vertices of 3D Shapes

Cube: 6 square faces, 12 edges, 8 vertices.
Cuboid: 6 rectangular faces (opposite faces equal), 12 edges, 8 vertices.
Triangular prism: 5 faces (2 triangles, 3 rectangles), 9 edges, 6 vertices.
Cylinder: 2 circular faces and 1 curved surface; 2 edges (circles), 0 vertices.
Cone: 1 circular face, 1 curved surface, 1 vertex (apex), 1 curved edge.
Sphere: 1 curved surface, 0 edges, 0 vertices.
Square-based pyramid: 5 faces (1 square, 4 triangles), 8 edges, 5 vertices.

Square-based pyramid

Prisms and Pyramids

A right prism has a constant polygonal cross-section and rectangular lateral faces. The face, edge and vertex formulas below apply to polygonal prisms.
Cuboid and cube are special prisms with rectangular cross-sections.
A cylinder has a constant circular cross-section, but it is not a polygonal prism.
Pyramid: a solid with a polygon base and triangular faces meeting at a common apex; named by its base (e.g., square-based pyramid).
A cone has a circular base and curved surface meeting at an apex; it is not a polygon-based pyramid.
For a prism, number of faces = (number of sides of cross-section) + 2; edges = 3 × (number of sides); vertices = 2 × (number of sides).
For a pyramid with an n-sided base: faces = n + 1, edges = 2n, vertices = n + 1.

Cylinder: flat and curved surfaces

Nets of 3D Shapes

A net is a 2D pattern that folds to form a 3D shape.
A cube has 11 different nets; each net has 6 squares arranged so they fold without overlap.
To check a net: ensure all faces are present and connected, and that folding along edges produces the correct solid.
Common nets: cuboid (6 rectangles), triangular prism (2 triangles + 3 rectangles), square-based pyramid (1 square + 4 triangles).
Cylinder net: two circles and a rectangle (curved surface unrolled).
Cone net: one circle and a sector of a circle.
A net must not have flaps that overlap when folded.

A valid cube net

Plans and Elevations

Plan: the view from directly above (top view).
Front elevation: the view from the front.
Side elevation: the view from the side (often right side).
For a 3D shape made of cubes, draw the outline of each view, showing visible squares.
Plan and elevations are 2D drawings that show the shape's dimensions from different directions.
Example: a cuboid of length 3, width 2, height 1 has plan 3×2, front elevation 3×1, side elevation 2×1.

Views of a cuboid

Common Misconceptions to Avoid

A square is a rectangle (it has 4 right angles and opposite sides equal) – don't think they are unrelated.
A rhombus with equal sides is not necessarily a square unless all angles are 90°.
Here, count flat faces separately from curved surfaces. The circular rims of cylinders and cones are curved edges.
When counting edges of a pyramid, remember the base edges plus the slant edges.
A net must fold without gaps or overlaps; a misplaced flap makes it invalid.
Don't confuse plan (top view) with front elevation (front view).

Typical Exam Questions

Name a shape from its properties (e.g., 'I have 4 equal sides and 4 right angles' → square).
State the number of faces, edges, and vertices of a given solid (e.g., a triangular prism).
Draw the net of a given solid (e.g., a cuboid).
Draw the plan and elevations of a 3D shape built from cubes.
Identify which of several nets forms a cube (only 11 of many arrangements work).
Use correct notation: mark parallel sides with arrows, equal sides with dashes, and right angles with squares.

Diapos

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Questions d'entraînement

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1.Which of the following is the correct notation for a point?
Easy
- AA
- BAB
- C∠ABC
- Dline l
2.How many edges does a triangular prism have?
Easy
- A6
- B9
- C12
- D5
3.A square is a type of rectangle.
Easy
True or false?
4.Which of the following is a regular polygon?
Medium
- AEquilateral triangle
- BNon-square rectangle
- CNon-equilateral isosceles triangle
- DNon-rectangular parallelogram
5.Which of the following are properties of a cube? (select all that apply)
Medium
- A6 faces
- B8 vertices
- C12 edges
- DAll faces are rectangles
- EAll faces are squares
6.Match each solid to its number of flat faces. Count curved surfaces separately.
Medium
- Cube
- Triangular prism
- Cylinder
- 6
- 5
- 2
7.Arrange the following shapes in order of increasing number of sides: pentagon, triangle, hexagon, quadrilateral.
Medium
- Triangle
- Quadrilateral
- Pentagon
- Hexagon
8.Match each solid to the correct number of edges.
Easy
- Cube
- Triangular prism
- Square-based pyramid
- Cylinder
- 12
- 9
- 8
- 2

Plus de questions

9.Which of the following are properties of a cone? (select all that apply)
Medium
- AIt has one circular base.
- BIt has a curved surface.
- CIt has a vertex (apex).
- DIt has two circular faces.
- EIt has a rectangular face.
10.A cylinder has exactly two flat faces.
Easy
True or false?
11.Which of the following is a correct net for a cube?
Medium
- AFour squares in a row, with one above and one below the second square
- BSix squares in a single straight row
- CSix squares in a 2 by 3 rectangle
- DFive squares arranged as a cross
12.A cuboid has dimensions 4 cm by 3 cm by 2 cm. How many faces does it have?
Medium
- A6
- B8
- C12
- D24
13.Match each term to its definition.
Medium
- Vertex
- Edge
- Face
- Plane
- A corner point where edges meet
- A line segment where two faces meet
- A flat surface
- A flat two-dimensional surface that extends infinitely
14.A solid has one square base and four triangular faces meeting at an apex. What is it called?
Medium
- ASquare-based pyramid
- BTriangular prism
- CCuboid
- DCone
15.Which of the following is the correct net for a cylinder?
Medium
- ATwo circles and a rectangle
- BOne circle and a triangle
- CTwo rectangles and a square
- DOne circle and a rectangle
16.Which of the following are properties of a regular hexagon? (select all that apply)
Medium
- AAll sides are equal
- BAll interior angles are equal
- CIt has 6 lines of symmetry
- DIt has 6 vertices
- EIt has 6 diagonals
17.Arrange the following shapes in order of increasing number of vertices.
Medium
- Triangle
- Square
- Pentagon
- Hexagon
18.A rectangular prism has dimensions 3 cm by 4 cm by 5 cm. Which calculation gives its volume in cm³?
Medium
- A3 + 4 + 5
- B3 × 4 × 5
- C2(3×4) + 2(3×5) + 2(4×5)
- D2(3+4+5)
19.A cylinder has a radius of 3 cm and a height of 5 cm. What is the volume in cm³? (Use π = 3.14)
Medium
- A141.3
- B47.1
- C94.2
- D28.26
20.A cone has a base radius of 4 cm and a height of 9 cm. What is the volume in cm³? (Use π = 3.14)
Medium
- A150.72
- B452.16
- C50.24
- D113.04
21.A sphere has a curved surface and no edges.
Easy
True or false?
22.A square-based pyramid has 5 faces and 8 edges.
Medium
True or false?
23.Which of the following are properties of a triangular prism? (select all that apply)
Medium
- AIt has 5 faces.
- BIt has 9 edges.
- CIt has 6 vertices.
- DIt has a curved surface.
24.Match each solid to the correct number of vertices.
Medium
- Cube
- Triangular prism
- Square-based pyramid
- 8
- 6
- 5
25.Arrange these solids in order of increasing number of flat faces: cylinder, tetrahedron, triangular prism and cube.
Medium
- Tetrahedron
- Cylinder
- Triangular prism
- Cube
26.A point is a location in space with no size.
Easy
True or false?
27.Which description explicitly uses the bar symbol for a line segment with endpoints A and B?
Easy
- AAB
- Bline AB
- CAB with a line above
- DA-B
28.Which of the following are properties of a cuboid? (select all that apply)
Easy
- A6 faces
- B8 vertices
- C12 edges
- DAll faces are squares
- EAll faces are rectangles
29.Match each polygon to the correct number of sides.
Medium
- Pentagon
- Hexagon
- Octagon
- 5
- 6
- 8
30.All squares are rectangles.
Easy
True or false?
31.A solid has six congruent square faces, twelve edges and eight vertices. What is its most specific name?
Medium
- ACuboid
- BCube
- CTriangular prism
- DSquare-based pyramid
32.Arrange the following solids in order of increasing number of edges: cylinder, triangular prism, cube, square-based pyramid.
Medium
- cylinder
- square-based pyramid
- triangular prism
- cube
33.Match each term to its correct definition.
Easy
- Parallel lines
- Perpendicular lines
- Right angle
- Plane
- Distinct lines in the same plane that never meet
- Lines that meet at 90°
- An angle of 90°
- A flat surface
34.Arrange these polygons in order of increasing number of sides: octagon, triangle, hexagon, quadrilateral.
Easy
- octagon
- triangle
- hexagon
- quadrilateral
35.Which of the following are properties of a regular pentagon? (select all that apply)
Medium
- AAll sides are equal
- BAll interior angles are equal
- CIt has 5 lines of symmetry
- DIt has 5 vertices
- EIt has 6 sides
36.A cylinder has two curved surfaces.
Easy
True or false?
37.Which of the following nets will NOT form a cube?
Medium
- AFour squares in a row, with one above and one below the second square
- BFour squares in a row, with one above and one below the third square
- CFour squares in a row, with one above the first square and one below the fourth
- DSix squares arranged as a 2 by 3 rectangle
38.Which of the following are properties of a square-based pyramid? (select all that apply)
Medium
- AIt has 5 faces
- BIt has 8 edges
- CIt has 5 vertices
- DIt has 4 triangular faces
- EIt has a square base
39.Arrange these solids in order of increasing number of flat faces: cylinder, tetrahedron, triangular prism and cube.
Medium
- cylinder
- triangular prism
- tetrahedron
- cube
40.A cuboid has dimensions 5 cm, 4 cm, and 3 cm. What is its volume in cm³?
Medium
- A12
- B27
- C60
- D120
41.How many vertices does a square-based pyramid have?
Medium
42.Complete the sentence with the correct term.
Easy
A line that intersects another line at a right angle is said to be ____ to it.
43.What is the name of the solid that has one circular base and a curved surface meeting at a point?
Easy
44.How many vertices does the cuboid have?
Medium
45.How many edges does the cuboid have?
Medium
46.How many faces does the cuboid have?
Medium
47.A cube has 12 edges. How many edges does a triangular prism have?
Medium
48.Fill in the blanks with the correct terms.
Easy
A polygon with all sides equal and all angles equal is called a ____ polygon.
49.Complete the sentence.
Medium
Two lines that are always the same distance apart and never meet are called ____ lines.
50.How many edges does a square-based pyramid have?
Medium
51.A cuboid has dimensions 5 cm by 4 cm by 3 cm. How many faces does it have?
Easy
52.A triangular prism has a triangular face with base 6 cm and height 4 cm. If the prism is 10 cm long, what is the volume in cm³?
Medium
53.Explain why a square is also a rectangle.
Medium
54.Explain the difference between a plan view and a front elevation of a 3D shape.
Medium
55.How many vertices does a square-based pyramid have?
Medium
- A4
- B7
- C6
- D5
56.How many vertices does the cuboid have?
Medium
- A9
- B8
- C7
- D10
57.How many edges does the cuboid have?
Medium
- A14
- B11
- C13
- D12
58.How many faces does the cuboid have?
Medium
- A7
- B5
- C8
- D6
59.A cube has 12 edges. How many edges does a triangular prism have?
Medium
- A11 edges
- B9 edges
- C8 edges
- D10 edges
60.How many edges does a square-based pyramid have?
Medium
- A9 edges
- B10 edges
- C8 edges
- D7 edges
61.A cuboid has dimensions 5 cm by 4 cm by 3 cm. How many faces does it have?
Easy
- A7
- B6
- C5
- D8
62.A triangular prism has a triangular face with base 6 cm and height 4 cm. If the prism is 10 cm long, what is the volume in cm³?
Medium
- A108 cm³
- B132 cm³
- C120 cm³
- D144 cm³

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