A Right Triangle Revolving Around a Side
Imagine a right triangle ABC with sides measuring 5 cm, 12 cm, and 13 cm. The longest side, 13 cm, is the hypotenuse, and the other two sides, 5 cm and 12 cm, form the legs of the triangle. Now, let's take this triangle and rotate it around the side that measures 12 cm. What shape do we get?
The Resulting Solid: A Cone
When we rotate the triangle around the 12 cm side, we create a cone.
- The base of the cone is a circle with a radius of 5 cm (the length of the other leg of the triangle).
- The height of the cone is 12 cm (the side we rotated around).
- The slant height of the cone is 13 cm (the hypotenuse of the original triangle).
Calculating the Volume and Surface Area
Now, let's find the volume and surface area of this cone.
1. Volume:
The volume (V) of a cone is given by the formula:
V = (1/3)πr²h
Where:
- r is the radius of the base (5 cm)
- h is the height of the cone (12 cm)
Plugging in the values:
V = (1/3) * π * (5 cm)² * 12 cm V = 100π cm³
2. Surface Area:
The surface area (SA) of a cone consists of two parts: the base and the lateral surface.
-
Area of the base: πr² = π * (5 cm)² = 25π cm²
-
Lateral surface area: πrl = π * 5 cm * 13 cm = 65π cm²
Therefore, the total surface area of the cone is:
SA = 25π cm² + 65π cm² = 90π cm²
Conclusion
By rotating a right triangle around one of its legs, we obtain a cone. We can then calculate the volume and surface area of the cone using the appropriate formulas. This exercise demonstrates how understanding geometric concepts can be used to solve real-world problems.
