REGULAR POLYGONS

 

Definition

A polygon with all sides and angles equal.

Examples

1. Equilateral Triangle (3 sides)

2. Square (4 sides)

3. Regular Pentagon (5 sides)

4. Regular Hexagon (6 sides)

Properties

- All interior angles are equal.

- All exterior angles are equal.

- Sum of interior angles = (n-2) × 180°

- Each interior angle = (n-2) × 180° / n

Uses

- Design and architecture

- Art and patterns

- Nature (e.g., honeycombs)

POLYGONS

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A polygon is any 2D shape that is closed and made up of straight lines. If it has curves or an open end, it’s not a polygon.

Characteristics of Polygons

• It is formed using three or more straight line segments.
• The line segments are called sides.
• The meeting point of two sides is called a vertex.
• Polygons have interior angles formed between the sides.
• The sides of a polygon do not intersect each other except at the vertices.
• Polygons have interior angles formed between the sides.
• The sides of a polygon do not intersect each other except at the vertices.
• The line segments are called sides.
• The meeting point of two sides is called a vertex.
• Polygons have interior angles formed between the sides.

Different Types of Polygons

Based on the number of sides:

• Triangle – 3 sides
• Quadrilateral – 4 sides
• Pentagon – 5 sides
• Hexagon – 6 sides
• Heptagon – 7 sides
• Octagon – 8 sides

Based on sides and angles:

• Regular Polygon – All sides and angles are equal.
• Irregular Polygon – Sides and angles are not equal.

SOLIDS

 

A mathematical solid is the portion of space enclosed by one or more surfaces. It is characterized by three measurements:

• Length
• Width (or Breadth)
• Height (or Depth)

DIFFERENT TYPES 

1. Cube

A three-dimensional solid with 6 square faces, 12 edges, and 8 vertices.

2. Cuboid

A three-dimensional solid with 6 rectangular faces, 12 edges, and 8 vertices.

3. Sphere

A three-dimensional solid with a curved surface, no edges, and no vertices.

4. Cylinder

A three-dimensional solid with 2 circular faces, 1 curved surface, and no vertices.

5. Cone

A three-dimensional solid with 1 circular face, 1 curved surface, and 1 vertex.

6. Pyramid

A three-dimensional solid with a polygonal base, triangular faces, and 1 vertex.

7. Prism

A three-dimensional solid with 2 identical polygonal faces, rectangular faces, and no vertices.

PARALLEL LINES

In geometry, parallel lines are two or more lines that reside in the same plane but never intersect, no matter how far they are extended in either direction.

Characteristics of Parallel Lines

• Parallel lines never meet or intersect.
• The distance between the two lines is always the same at every point.
• They lie in the same plane (coplanar).
• When a transversal cuts two parallel lines, several special angle relationships are formed.

Angles Formed by a Transversal

When a transversal intersects two parallel lines, the following angles are formed:

Corresponding angles – equal in measure.

Alternate interior angles – equal in measure.

Alternate exterior angles – equal in measure.

Co-interior (consecutive interior) angles – their sum is 180°.

Co-interior (consecutive interior) angles – their sum is 180°.

TRIANGLES

DEFINITION 

Triangle is a three-sided polygon. It is one of the most fundamental shapes in the geometry.

TYPES OF TRIANGLES 

1. Equilateral Triangle

A triangle with 3 equal sides and 3 equal angles (60° each).

2. Isosceles Triangle

A triangle with 2 equal sides and 2 equal angles.

3. Scalene Triangle

A triangle with all sides and angles unequal.

4. Right-Angled Triangle

A triangle with 1 right angle (90°).

5. Obtuse Triangle

A triangle with 1 obtuse angle (> 90°).

6. Acute Triangle

A triangle with all angles acute (< 90°).

QUADRILATERALS

DEFINITION

 A quadrilateral is a fundamental polygon defined by four sides, four vertices, and four angles. The name itself is a giveaway, derived from the Latin quadri (four) and latus (side).

To qualify as a quadrilateral, a shape must be a closed, two-dimensional figure made of straight line segments.

KEY CHARACTERISTICS 

Regardless of their specific name (like square or trapezoid), all quadrilaterals share these essential traits:

Sides: 4 straight edges.

Vertices: 4 corners where the sides meet.

Interior Angles: The sum of all four interior angles is always 360°.

Diagonals: Every quadrilateral has exactly 2 diagonals (lines connecting opposite vertices).

DIFFERENT TYPES 

1. Square

A quadrilateral with 4 equal sides and 4 right angles.

2. Rectangle

A quadrilateral with 4 right angles and opposite sides of equal length.

3. Rhombus

A quadrilateral with 4 equal sides and opposite angles equal.

4. Trapezium

A quadrilateral with 1 pair of parallel sides.

5. Parallelogram

A quadrilateral with opposite sides parallel and equal.

6. Kite

A quadrilateral with 2 pairs of adjacent sides equal.