Geometry - AxomExpress

Geometry Notes for Class V to X

📘 Geometry Contents

1. Basic Geometrical Ideas 2. Lines and Angles 3. Triangles 4. Quadrilaterals 5. Circle 6. Polygons 7. Symmetry 8. Important Geometry Formulas 9. Mensuration 10. Coordinate Geometry 11. Geometrical Construction 12. Theorems 13. Surface Area and Volume 14. Practical Geometry Tips

1. Basic Geometrical Ideas

Geometry is the branch of mathematics that deals with shapes, sizes, positions and measurements.

Point

A point shows an exact location. It has no length or breadth.

Line

A line extends endlessly in both directions.

Line Segment

A line segment has two fixed end points.

A line segment AB has fixed length.

2. Lines and Angles

Angle

An angle is formed when two rays meet at one point.

Types of Angles

Acute Angle → less than 90°
Right Angle → exactly 90°
Obtuse Angle → greater than 90° but less than 180°
Straight Angle → 180°

Example: A clock showing 3 o'clock forms a right angle.

3. Triangles

A triangle has 3 sides and 3 angles.

Types of Triangles by Sides

Equilateral Triangle → all sides equal
Isosceles Triangle → two sides equal
Scalene Triangle → all sides different

Types by Angles

Acute Triangle
Right Triangle
Obtuse Triangle

Sum of all interior angles of a triangle = 180°

If two angles are 50° and 60°, then third angle = 180° − (50° + 60°) = 70°.

4. Quadrilaterals

A quadrilateral has four sides.

Types of Quadrilaterals

Square
Rectangle
Parallelogram
Rhombus
Trapezium

A rectangle has opposite sides equal and all angles are 90°.

5. Circle

A circle is a round closed figure where every point is equally distant from the center.

Important Terms

Radius → distance from center to circle
Diameter → longest chord through center
Circumference → boundary of circle

Diameter = 2 × Radius

6. Polygons

A polygon is a closed shape made of line segments.

Triangle → 3 sides
Quadrilateral → 4 sides
Pentagon → 5 sides
Hexagon → 6 sides
Octagon → 8 sides

Regular polygons have all sides equal.

7. Symmetry

A figure is symmetrical if it can be divided into two equal halves.

The dotted line is called the line of symmetry.

📐 Complete Geometry Formulas with Explanation & Diagrams

Each formula below contains:

Meaning of formula
Simple explanation
Diagram
Worked example

1. Area of Rectangle

Formula: A = l × w

The area of rectangle is found by multiplying length and width.

Length = 5 cm
Width = 3 cm
Area = 5 × 3 = 15 cm²

2. Perimeter of Rectangle

Formula: P = 2(l + w)

Perimeter means total boundary length of rectangle.

Length = 5 cm
Width = 3 cm
P = 2(5 + 3) = 16 cm

3. Area of Square

Formula: A = s²

Multiply side by itself.

Side = 4 cm
Area = 4² = 16 cm²

4. Perimeter of Square

Formula: P = 4s

Since all sides are equal, multiply one side by 4.

Side = 4 cm
Perimeter = 4 × 4 = 16 cm

5. Area of Triangle

Formula: A = 1/2 × b × h

Multiply base and height, then divide by 2.

Base = 6 cm
Height = 4 cm
Area = 1/2 × 6 × 4 = 12 cm²

6. Heron's Formula

Formula: A = √[s(s-a)(s-b)(s-c)]

Used when all three sides are known.

s = (a+b+c)/2

Sides = 3 cm, 4 cm, 5 cm
s = (3+4+5)/2 = 6
Area = √[6(6−3)(6−4)(6−5)]
Area = 6 cm²

7. Circumference of Circle

Formula: C = 2πr

Circumference means boundary length of circle.

Radius = 3 cm
C = 2π × 3 = 6π cm

8. Area of Circle

Formula: A = πr²

Area covered inside the circle.

Radius = 3 cm
Area = π × 3²
Area = 9π cm²

9. Volume of Sphere

Formula: V = 4/3 πr³

Measures space inside a sphere.

Radius = 2 cm
V = 4/3 × π × 8
V = 32/3 π cm³

10. Surface Area of Sphere

Formula: A = 4πr²

Total outer surface area of sphere.

Radius = 2 cm
A = 4π × 4
A = 16π cm²

11. Volume of Cylinder

Formula: V = πr²h

Area of circular base multiplied by height.

Radius = 2 cm
Height = 5 cm
V = π × 4 × 5 = 20π cm³

12. Volume of Cone

Formula: V = 1/3 πr²h

Volume of cone is one-third of cylinder volume.

Radius = 2 cm
Height = 6 cm
V = 1/3 × π × 4 × 6
V = 8π cm³

13. Area of Trapezium

Formula: A = 1/2(a+b)h

Add parallel sides and multiply by height.

Parallel sides = 3 cm and 5 cm
Height = 4 cm
Area = 1/2 × (3+5) × 4 = 16 cm²

14. Area of Parallelogram

Formula: A = bh

Multiply base and perpendicular height.

Base = 5 cm
Height = 3 cm
Area = 15 cm²

15. Area of Sector

Formula: A = 1/2 r²θ

Sector is a slice of circle.

Radius = 2 cm
θ = π/2
Area = 1/2 × 4 × π/2 = π cm²

16. Arc Length

Formula: L = rθ

Arc length means curved distance of sector.

Radius = 3 cm
θ = π/3
L = 3 × π/3 = π cm

π (Pi) ≈ 3.14
Area → square units (cm²)
Volume → cubic units (cm³)

8. Mensuration

Mensuration deals with area, perimeter and volume.

Perimeter of Rectangle

Perimeter = 2 × (Length + Breadth)

Area of Rectangle

Area = Length × Breadth

Area of Triangle

Area = 1/2 × Base × Height

If length = 8 cm and breadth = 5 cm, Perimeter = 2 × (8 + 5) = 26 cm.

9. Coordinate Geometry

Coordinate geometry helps us locate points on a graph.

Horizontal line = X-axis Vertical line = Y-axis

10. Geometrical Construction

Construction means drawing figures accurately using compass, ruler and protractor.

Important Constructions

Drawing a line segment
Drawing angles
Constructing triangles
Drawing perpendicular bisector

To draw 60° angle:

Draw a ray.
Place the protractor.
Mark 60°.
Join the point.

11. Important Theorems

Pythagoras Theorem

In a right triangle:

Hypotenuse² = Base² + Perpendicular²

If base = 3 cm and height = 4 cm, Hypotenuse = √(3² + 4²) = 5 cm.

12. Surface Area and Volume

Cube

Surface Area = 6a²
Volume = a³

Cuboid

Volume = Length × Breadth × Height

13. Practical Geometry Tips

Always use sharp pencil for construction.
Measure angles carefully.
Practice drawing figures neatly.
Remember important formulas daily.
Understand diagrams instead of memorizing only.

Geometry becomes easy when you practice drawing and solving regularly.