# Mathematics Project on Pythagoras Theorem and its Extension

## Pythagoras Theorem and its Extension

Objective

To understand the Pythagoras theorem using geometrical representation by using areas of squares on each side of a right triangle, and extending it to three dimensional objects using volumes.

Pythagoras Theorem states that square on Hypotenuse of a right triangle is equal to sum of squares on remaining two sides.

1. For a Right Triangle

Description

**1. **Cut a triangle of sides 6cm, 8cm and 10cm

**2. **Cut squares equal to sides of triangle.

**3. **Divided each square into small squares of 1cm each

Calculations

1. The number of 1 cm squares in square drawn on Side of 6 cm were 36

2. The number of 1 cm squares in square drawn on Side of 8 cm were 64

3. Sum of square on these two sides = 64 + 36 = 100

4. The number of 1 cm squares in square drawn on Side of 10 cm (hypotenuse) were 100

5. .*. square on Hypotenuse of a right triangle is equal to

sum of squares on remaining two sides.

Hence Pythagoras Theorem is verified for a right triangle

2. For Right Circular Cylinder Description

**1. **Took right circular cylinders of radii 6cm, 8cm and 10cm.

**2. **Filled the two smaller cylinders (r = 6cm, 8cm) with sand.

**3. **Keep the cylinder with r = 10 cm empty.

Method

1. Poured the sand from cylinders with radii 6cm and 8cm into the biggest cylinder (r = 10cm).

**4. **We found that the bigger cylinder is completely filled with sand.

**5. **This shows volume of cylinder with radius 10cm = sum of volumes of the cylinders with volume 6cm and 8cm.

Hence Pythagoras Theorem can be extended for right

circular cylinders

3. For Right Circular Cone Description

**1. **Took right circular cone of radii 6cm, 8cm and 10cm.

**2. **Filled the two smaller cones (r = 6cm, 8cm) with sand.

**3. **Keep the cone with r = 10cm empty.

Method:

1. Poured the sand from cone with radii 6cm and 8cm into the biggest cone (r = 10cm).

**4. **We find that the bigger cone is completely filled with sand.

**5. **This shows volume of cone with radius 10cm = sum of volumes of the cone with volume 6 cm and 8cm.

Hence Pythagoras Theorem can be extended for right

circular cone

Observation :- I observed that the Pythagoras theorem is true for right triangles and can be extended for three dimensional figures such as cylinders and cones.