Mathematics Project on Pythagoras Theorem and its Extension

Mathematics Project on 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 the square on the hypotenuse of a right triangle is equal to the sum of squares on the remaining two sides.

1. For a Right Triangle

Description:

1. Cut a triangle of sides 6 cm, 8 cm and 10 cm.
2. Cut squares equal to the sides of the triangle.
3. Divide each square into small squares of 1 cm each.

Calculations:

1. The number of 1 cm squares in the square drawn on the side of 6 cm were 36.
2. The number of 1 cm squares in the square drawn on the side of 8 cm were 64.
3. Sum of squares on these two sides = 64 + 36 = 100.
4. The number of 1 cm squares in the square drawn on the side of 10 cm (hypotenuse) were 100.
5. Therefore, the square on the hypotenuse of a right triangle is equal to the sum of squares on the 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 6 cm, 8 cm and 10 cm.
2. Filled the two smaller cylinders (r = 6 cm, r = 8 cm) with sand.
3. Kept the cylinder with r = 10 cm empty.

Method:

1. Poured the sand from cylinders with radii 6 cm and 8 cm into the biggest cylinder (r = 10 cm).
2. We found that the bigger cylinder is completely filled with sand.
3. This shows volume of cylinder with radius 10 cm = sum of volumes of the cylinders with radii 6 cm and 8 cm.

Hence Pythagoras Theorem can be extended for right circular cylinders.

3. For Right Circular Cone

Description:

1. Took right circular cones of radii 6 cm, 8 cm and 10 cm.
2. Filled the two smaller cones (r = 6 cm, r = 8 cm) with sand.
3. Kept the cone with r = 10 cm empty.

Method:

1. Poured the sand from cones with radii 6 cm and 8 cm into the biggest cone (r = 10 cm).
2. We found that the bigger cone is completely filled with sand.
3. This shows volume of cone with radius 10 cm = sum of volumes of the cones with radii 6 cm and 8 cm.

Hence Pythagoras Theorem can be extended for right circular cones.

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.