Basic Math

Geometry | Third Grade

📐 Geometry - Grade 3

Understanding Geometry!

Geometry is the study of shapes, sizes, positions, and properties of things around us! Let's explore 2D shapes, 3D shapes, and more!

📏 Identify Two-Dimensional Shapes

What are 2D Shapes?

Two-dimensional (2D) shapes are flat shapes that have only two dimensions: length and width. They have no thickness or depth!

2D shapes are also called flat shapes or plane shapes.

Common 2D Shapes:

Shape NameNumber of SidesProperties
Circle ⭕0 (curved)Round, no corners
Triangle 🔺33 sides, 3 corners
Square 🟦44 equal sides, 4 right angles
Rectangle 🟥4Opposite sides equal, 4 right angles
Pentagon55 sides, 5 corners
Hexagon66 sides, 6 corners
Octagon88 sides, 8 corners (like STOP sign)

🔢 Count and Compare Sides and Vertices

Important Terms:

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

Counting Rule:

Number of Sides = Number of Vertices = Number of Angles
(For all polygons)

Examples:

Triangle: 3 sides, 3 vertices, 3 angles
Square: 4 sides, 4 vertices, 4 angles
Pentagon: 5 sides, 5 vertices, 5 angles
Hexagon: 6 sides, 6 vertices, 6 angles
Octagon: 8 sides, 8 vertices, 8 angles

💡 Tip: To count sides, trace around the shape. To count vertices, count the corners!

📦 Identify Three-Dimensional Shapes

What are 3D Shapes?

Three-dimensional (3D) shapes are solid shapes that have three dimensions: length, width, and height (depth)!

3D shapes are also called solid shapes. You can hold them in your hand!

Common 3D Shapes:

Shape NameFacesEdgesVerticesExample
Sphere1 curved00Ball, globe
Cube6128Dice, Rubik's cube
Cuboid6128Box, book
Cylinder3 (2 flat, 1 curved)20Can, pipe
Cone2 (1 flat, 1 curved)11Ice cream cone, party hat
Pyramid585Egyptian pyramid, tent

🔢 Count Vertices, Edges and Faces

Important 3D Terms:

Face: A flat surface of a 3D shape
Edge: A line where two faces meet
Vertex (Vertices): A corner where edges meet

Euler's Formula for 3D Shapes:

\(\text{Faces} + \text{Vertices} = \text{Edges} + 2\)

\(F + V = E + 2\)

Example: Cube

Faces: 6 (all squares)
Vertices: 8 (corners)
Edges: 12 (lines where faces meet)

Check with Euler's Formula:
\(6 + 8 = 12 + 2\)
\(14 = 14\) ✓

🎲 Identify Faces of Three-Dimensional Shapes

What are Faces?

Faces are the flat or curved surfaces of a 3D shape. Each face is a 2D shape!

Examples:

Cube:

All 6 faces are squares

Cuboid (Rectangular Prism):

All 6 faces are rectangles

Cylinder:

2 faces are circles (top and bottom)
1 face is curved (side)

Square Pyramid:

1 face is a square (base)
4 faces are triangles (sides)

⬡ Is It a Polygon?

What is a Polygon?

A polygon is a closed 2D shape made up of straight line segments!

Rules for a Polygon:

  1. Must be CLOSED - All sides connect
  2. Must have STRAIGHT sides - No curves
  3. Must have at least 3 sides - Minimum is a triangle
  4. Sides can only touch at vertices - Lines don't cross

Polygons vs Non-Polygons:

Polygons ✓NOT Polygons ✗
Triangle, Square, RectangleCircle (curved)
Pentagon, Hexagon, OctagonOval (curved)
All sides are straightOpen shapes (not closed)
Completely closedShapes with curved sides

🔄 Reflection, Rotation and Translation

What are Transformations?

Transformations are ways to move or change the position of a shape!

Three Types of Transformations:

1. Reflection (Flip) 🪞

Reflection means flipping a shape over a line (like a mirror)!

• The shape stays the same size
• It flips to the opposite side
• Like looking in a mirror
• Example: Flip a letter "b" to get "d"

2. Rotation (Turn) 🔄

Rotation means turning a shape around a point!

• The shape stays the same size
• It turns around a center point
• Can turn 90°, 180°, 270°, or 360°
• Example: Turn a square 90° (quarter turn)

3. Translation (Slide) ➡️

Translation means sliding a shape to a new position!

• The shape stays the same size
• It moves in a straight line
• Can move up, down, left, or right
• Example: Slide a triangle 3 units right

💡 Remember: All transformations keep the shape the same size and form!

🦋 Symmetry

What is Symmetry?

A shape has symmetry when one half is a mirror image of the other half!

The imaginary line that divides the shape into two equal halves is called the line of symmetry.

Lines of Symmetry in Shapes:

ShapeLines of Symmetry
CircleInfinite (countless)
Square4
Rectangle2
Equilateral Triangle3
Regular Pentagon5
Regular Hexagon6

How to Test for Symmetry:

  1. Fold the shape in half (or imagine folding)
  2. Check if both halves match perfectly
  3. If they match, it has symmetry!
  4. The fold line is the line of symmetry

🗺️ Maps

What are Maps in Geometry?

In geometry, maps help us locate shapes and objects using coordinates or directions!

Grid Coordinates:

Maps use a grid with rows and columns to show where things are located!

Horizontal axis (x-axis): Goes left to right
Vertical axis (y-axis): Goes up and down
Coordinates: Written as (x, y)
Origin: The point (0, 0) where axes meet

Directions on Maps:

North ⬆️ - Up
South ⬇️ - Down
East ➡️ - Right
West ⬅️ - Left

📐 Find the Area of Rectangles and Squares

What is Area?

Area is the amount of space inside a 2D shape! It tells us how many square units fit inside.

Area is measured in square units like cm², m², or square inches.

Area of a Rectangle:

\(\text{Area} = \text{Length} \times \text{Width}\)

\(A = l \times w\)

Example:

A rectangle has length = 8 cm and width = 5 cm. Find the area.

Formula: \(A = l \times w\)
Substitute: \(A = 8 \times 5\)
Calculate: \(A = 40\)
Answer: 40 cm² ✓

Area of a Square:

\(\text{Area} = \text{Side} \times \text{Side}\)

\(A = s \times s = s^2\)

Example:

A square has side = 6 cm. Find the area.

Formula: \(A = s^2\)
Substitute: \(A = 6^2\)
Calculate: \(A = 6 \times 6 = 36\)
Answer: 36 cm² ✓

💡 Remember: Always write the answer in square units (cm², m², etc.)!

❓ Find the Missing Side Length of a Rectangle

How to Find Missing Side Length:

If you know the area and one side, you can find the missing side!

Formulas:

If you know Area and Length:
\(\text{Width} = \text{Area} \div \text{Length}\)
\(w = A \div l\)

If you know Area and Width:
\(\text{Length} = \text{Area} \div \text{Width}\)
\(l = A \div w\)

Examples:

Example 1: Find the Width

A rectangle has area = 48 cm² and length = 8 cm. Find the width.

Given: Area = 48 cm², Length = 8 cm
Formula: \(w = A \div l\)
Substitute: \(w = 48 \div 8\)
Calculate: \(w = 6\)
Answer: Width = 6 cm ✓

Example 2: Find the Length

A rectangle has area = 56 m² and width = 7 m. Find the length.

Given: Area = 56 m², Width = 7 m
Formula: \(l = A \div w\)
Substitute: \(l = 56 \div 7\)
Calculate: \(l = 8\)
Answer: Length = 8 m ✓

💡 Check Your Answer: Multiply length × width. You should get the area!

📝 Important Formulas Summary

2D Shapes:

Sides = Vertices = Angles
(For all polygons)

3D Shapes (Euler's Formula):

\(F + V = E + 2\)
(Faces + Vertices = Edges + 2)

Area Formulas:

Rectangle: \(A = l \times w\)

Square: \(A = s^2\)

Find Missing Side:
\(l = A \div w\) or \(w = A \div l\)

💡 Quick Learning Tips

  • 2D shapes are flat (length and width only)
  • 3D shapes are solid (length, width, and height)
  • Polygons have straight sides and are closed
  • Circles and ovals are NOT polygons (curved sides)
  • For 2D shapes: Sides = Vertices = Angles
  • For 3D shapes: F + V = E + 2 (Euler's Formula)
  • Face = flat surface, Edge = line where faces meet, Vertex = corner
  • Reflection = Flip, Rotation = Turn, Translation = Slide
  • Line of symmetry divides a shape into two equal mirror halves
  • Area of rectangle = length × width
  • Area of square = side × side (s²)
  • To find missing side: divide area by known side
  • Always use square units for area (cm², m², etc.)
  • A square is a special rectangle with all sides equal
  • Practice identifying shapes in real life!
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