A Right Triangle ABC with Sides 5, 12, and 13 is Revolved
A right triangle ABC with sides 5, 12, and 13 is a special type of triangle known as a Pythagorean triple. This means the sides satisfy the Pythagorean theorem: a² + b² = c², where 'c' is the hypotenuse.
When this triangle is revolved around different sides, it generates various three-dimensional shapes:
Revolving Around the Hypotenuse (Side 13)
Revolving the triangle around the hypotenuse (side 13) results in a cone.
- Base radius: Half the length of the shorter leg (5/2).
- Height: Half the length of the longer leg (12/2).
- Slant height: The hypotenuse (13).
Revolving Around the Shorter Leg (Side 5)
Revolving around the shorter leg (side 5) generates a right circular cone.
- Base radius: The length of the longer leg (12).
- Height: The length of the shorter leg (5).
- Slant height: The hypotenuse (13).
Revolving Around the Longer Leg (Side 12)
Rotating the triangle around the longer leg (side 12) results in a right circular cone.
- Base radius: The length of the shorter leg (5).
- Height: The length of the longer leg (12).
- Slant height: The hypotenuse (13).
Calculating the Volume and Surface Area
The volume and surface area of each generated shape can be calculated using the following formulas:
Cone:
- Volume: (1/3) * π * r² * h
- Surface Area: π * r * (r + l)
Where:
- r = radius of the base
- h = height of the cone
- l = slant height of the cone
By substituting the appropriate values from each case, the volume and surface area of each cone can be determined.
Conclusion
Revolving a right triangle with sides 5, 12, and 13 around different sides generates three distinct right circular cones. The specific dimensions of each cone depend on the side around which the revolution is performed. This exploration demonstrates the fascinating relationship between geometry and solid shapes.
