Geometry | Fourth Grade

Complete Notes & Formulas

1. Two-Dimensional (2D) Figures

Definition: 2D shapes are flat figures that have only two dimensions: length and width (no depth). They exist on a plane and have only area, not volume.

📐 Common 2D Shapes:

Shape	Sides	Properties
Circle	0	Round, no corners, all points equal distance from center
Triangle	3	3 sides, 3 vertices, 3 angles
Square	4	4 equal sides, 4 right angles, 4 vertices
Rectangle	4	Opposite sides equal, 4 right angles
Pentagon	5	5 sides, 5 vertices, 5 angles
Hexagon	6	6 sides, 6 vertices, 6 angles
Octagon	8	8 sides, 8 vertices, 8 angles

🔑 Key Terms:

Vertex (plural: Vertices): A corner where two sides meet
Side: A straight line segment forming part of the shape
Angle: The space between two sides that meet at a vertex

2. Three-Dimensional (3D) Figures

Definition: 3D shapes are solid figures that have three dimensions: length, width, and depth (or height). They occupy space and have volume.

🎲 Common 3D Shapes:

3D Shape	Description	Real-Life Example
Cube	6 square faces, all equal	Dice, Rubik's cube
Cuboid	6 rectangular faces	Shoe box, book
Sphere	Perfectly round, curved surface	Ball, globe
Cylinder	2 circular faces, 1 curved surface	Can, tube
Cone	1 circular base, 1 curved surface, 1 vertex	Ice cream cone, party hat
Pyramid	Polygon base, triangular faces meeting at apex	Egyptian pyramid, tent

3. Count Vertices, Edges, and Faces

Definition: 3D shapes have three important features: faces (flat or curved surfaces), edges (lines where faces meet), and vertices (corners where edges meet).

📐 Key Definitions:

Face:

A flat or curved surface of a 3D shape

Example: A cube has 6 square faces

Edge:

A line segment where two faces meet

Example: A cube has 12 edges

Vertex (Vertices - plural):

A point where two or more edges meet (corner)

Example: A cube has 8 vertices

📊 Faces, Edges, and Vertices Table:

3D Shape	Faces (F)	Edges (E)	Vertices (V)
Cube	6	12	8
Cuboid	6	12	8
Square Pyramid	5	8	5
Triangular Prism	5	9	6
Cylinder	3 (2 flat + 1 curved)	2	0
Cone	2 (1 flat + 1 curved)	1	1
Sphere	1 (curved)	0	0

🔑 Euler's Formula (for polyhedra):

F + V = E + 2

Faces + Vertices = Edges + 2

Example: Cube → 6 + 8 = 12 + 2 → 14 = 14 ✓

4-5. Identify Faces & Describe 3D Figures

Definition: The faces of 3D shapes can be identified by their 2D shape. Describing 3D figures involves naming the types and number of faces, edges, and vertices.

📦 Faces of Common 3D Shapes:

Cube:

• 6 faces - all are SQUARES

• All faces are identical (same size)

Cuboid (Rectangular Prism):

• 6 faces - all are RECTANGLES

• Opposite faces are identical

Square Pyramid:

• 5 faces: 1 SQUARE base + 4 TRIANGULAR faces

Triangular Prism:

• 5 faces: 2 TRIANGULAR ends + 3 RECTANGULAR sides

Cylinder:

• 3 surfaces: 2 CIRCULAR ends + 1 curved rectangular surface

6. Nets of Three-Dimensional Figures

Definition: A net is a 2D pattern that can be folded to make a 3D shape. When you unfold a 3D shape completely flat, you get its net.

📐 How Nets Work:

A net shows all the faces of a 3D shape laid flat
When folded along the edges, the net forms the 3D shape
The same 3D shape can have different nets
Not all 2D patterns can form 3D shapes

📦 Common Nets:

Cube Net:

• Made of 6 connected squares

• Can have 11 different net patterns

• Most common: Cross shape (+)

Cuboid Net:

• Made of 6 connected rectangles

• Opposite faces are identical

Pyramid Net:

• Square base with 4 triangles attached to each side

• Looks like a square with triangular flaps

💡 Tips to Identify Nets:

Count the faces - must match the 3D shape
Check the shape of each face
Imagine folding the net - will edges meet?
No overlapping when folded

7. Number of Sides in Polygons

Definition: A polygon is a closed 2D shape made of straight lines. Polygons are named based on the number of sides they have.

📐 Polygon Names by Sides:

Number of Sides	Polygon Name	Example
3	Triangle	Road signs, pyramids
4	Quadrilateral	Square, rectangle, trapezoid
5	Pentagon	Pentagon building
6	Hexagon	Honeycomb, nut
7	Heptagon	Some coins
8	Octagon	Stop sign
9	Nonagon	Some buildings
10	Decagon	Some coins

🔑 Key Formula:

Number of Sides = Number of Vertices = Number of Angles

In any polygon, sides, vertices, and angles are always equal in number

8. Identify Lines of Symmetry

Definition: A line of symmetry (or axis of symmetry) divides a shape into two identical halves that are mirror images of each other. When folded along this line, both halves match perfectly.

📐 Lines of Symmetry in Common Shapes:

Shape	Number of Lines of Symmetry	Description
Circle	Infinite (∞)	Any line through center
Square	4	2 diagonal, 2 through midpoints
Rectangle	2	Vertical and horizontal through center
Equilateral Triangle	3	From each vertex to opposite side
Isosceles Triangle	1	From apex to base midpoint
Regular Pentagon	5	From each vertex through center
Regular Hexagon	6	3 through vertices, 3 through sides

📝 How to Find Lines of Symmetry:

Imagine folding the shape
If both halves match exactly, it's a line of symmetry
Try vertical, horizontal, and diagonal lines
Count all possible lines of symmetry

🔑 Key Rule for Regular Polygons:

Lines of Symmetry = Number of Sides

Regular polygon with n sides has n lines of symmetry

9. Rotational Symmetry

Definition: A shape has rotational symmetry if it looks exactly the same after being rotated (turned) around a central point by less than 360°. The order of rotational symmetry is how many times it looks identical during one full rotation.

📐 Key Terms:

Center of Rotation: The fixed point around which the shape rotates
Order of Rotation: Number of times shape looks the same in 360° rotation
Angle of Rotation: Smallest angle needed to rotate shape to match itself

🔑 Key Formula:

Angle of Rotation = 360° ÷ Order of Rotation

📊 Rotational Symmetry Examples:

Shape	Order of Rotation	Angle of Rotation
Circle	Infinite (∞)	Any angle
Square	4	90°
Rectangle	2	180°
Equilateral Triangle	3	120°
Regular Pentagon	5	72°
Regular Hexagon	6	60°

💡 Important Notes:

All shapes have at least Order 1 (original position)
If order = 1, shape has NO rotational symmetry
Regular polygons: Order = number of sides
Shapes with rotational symmetry often have line symmetry too

Geometry Quick Reference Chart

Concept	Key Formula/Rule
2D Shapes	Flat, have length & width only
3D Shapes	Solid, have length, width & depth
Euler's Formula	F + V = E + 2
Polygon Sides	Sides = Vertices = Angles
Regular Polygon Symmetry	Lines of Symmetry = Number of Sides
Rotation Angle	360° ÷ Order of Rotation
Net	2D pattern that folds into 3D shape