1. What is Geometry?

Geometry is the study of shapes, sizes, positions, and properties of space.

It answers questions like:

How big is something? (area, volume)

How far apart are things? (distance)

What shape is it? (triangle, circle, etc.)

2. Basic Elements of Geometry

Point

A location with no size

Represented by a dot ()

Line

A straight path that extends infinitely in both directions

Line Segment

A part of a line with two endpoints

Ray

A line that starts at one point and extends forever in one direction

Plane

A flat 2D surface that extends infinitely

3. Angles

An angle is formed when two lines meet.

Types of Angles:

Acute: less than 90

Right: exactly 90

Obtuse: between 90 and 180

Straight: 180

Reflex: more than 180

4. Polygons (2D Shapes)

A polygon is a closed shape made of straight lines.

Common Polygons:

Triangle (3 sides)

Quadrilateral (4 sides)

Pentagon (5 sides)

Hexagon (6 sides)

Triangles

Types by sides:

Equilateral: all sides equal

Isosceles: two sides equal

Scalene: all sides different

Types by angles:

Acute triangle

Right triangle

Obtuse triangle

Key Formula:

Sum of angles = 180

Quadrilaterals

Square: all sides equal, all angles 90

Rectangle: opposite sides equal, all angles 90

Parallelogram: opposite sides parallel

Trapezoid: one pair of parallel sides

5. Circles

A circle is a set of points equally distant from a center.

Important Parts:

Radius (r): center to edge

Diameter (d): across the circle d = 2r

Circumference: distance around

Formulas:

Circumference = 2r

Area = r

6. Perimeter, Area, Volume

Perimeter

Distance around a shape

Area

Space inside a 2D shape

Examples:

Rectangle = length width

Triangle = base height

Volume

Space inside a 3D object

Examples:

Cube = side

Rectangular prism = l w h

Cylinder = rh

7. 3D Shapes (Solid Geometry)

Cube

Sphere

Cylinder

Cone

Pyramid

Each has:

Faces (flat surfaces)

Edges (lines)

Vertices (corners)

8. Coordinate Geometry

Uses a grid (x, y axes).

Distance Formula:

=

(

2

1

)

2

+

(

2

1

)

2

d=

(x

2

x

1

)

2

+(y

2

y

1

)

2

Midpoint Formula:

(

1

+

2

2

,

1

+

2

2

)

(

2

x

1

+x

2

,

2

y

1

+y

2

)

9. Transformations

How shapes move:

Translation: slide

Rotation: turn

Reflection: flip

Dilation: resize

10. Congruence & Similarity

Congruent:

Same size and shape

Similar:

Same shape, different size

11. Theorems & Important Rules

Pythagorean Theorem

For right triangles:

2

+

2

=

2

a

2

+b

2

=c

2

Parallel Lines

Alternate interior angles are equal

Corresponding angles are equal

12. Trigonometry (Part of Geometry)

Deals with triangles and angles.

Basic ratios:

sin = opposite/hypotenuse

cos = adjacent/hypotenuse

tan = opposite/adjacent

13. Advanced Geometry Fields

Euclidean Geometry

Flat space (what you usually learn in school)

Non-Euclidean Geometry

Curved space:

Spherical geometry

Hyperbolic geometry

Analytic Geometry

Uses algebra + geometry

Differential Geometry

Curves and surfaces (used in physics)

Topology

Study of shapes that can stretch without tearing

14. Why Geometry Matters

Used in:

Architecture

Engineering

Computer graphics

Astronomy

Navigation

Simple Summary

Geometry studies:

Shapes (triangle, circle)

Space (2D and 3D)

Measurements (area, volume)

Relationships (angles, lines)

If you want, I can:

Give easy-to-understand examples

Provide practice problems with answers

Or explain one topic deeply (like triangles or circles)

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