Mathematics Project on Pythagoras Theorem and its Extension

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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.

Name________ Class_________ Roll Number___ 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.

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