If S denotes the area of the curved surface of a right circular cone of height h and semivertical angle `alpha` then S equals

Options

A. `pi h^(2) tan^(2) alpha`
B. `(1)/(3) alphah^(2) tan^(2) alpha`
C. `pi h^(2) sec alpha tan alpha`
D. `(1)/(3) pi h^(2) sec alpha tan alpha`

Answer

To find the area of the curved surface \( S \) of a right circular cone with height \( h \) and semi-vertical angle \( \alpha \), we can follow these steps:

### Step 1: Understand the Geometry of the Cone
The right circular cone has a height \( h \) and a semi-vertical angle \( \alpha \). The radius \( r \) of the base of the cone can be expressed in terms of the height and the angle.

### Step 2: Relate the Radius to Height and Angle
Using the definition of the tangent function in a right triangle formed by the height, radius, and slant height:
\[ \tan(\alpha) = \frac{r}{h} \]
From this, we can express the radius \( r \) as:
\[ r = h \tan(\alpha) \]

### Step 3: Find the Slant Height
The slant height \( l \) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the expression for \( r \):
\[ l = \sqrt{h^2 + (h \tan(\alpha))^2} \]
\[ l = \sqrt{h^2 + h^2 \tan^2(\alpha)} = \sqrt{h^2(1 + \tan^2(\alpha))} \]
Using the identity \( 1 + \tan^2(\alpha) = \sec^2(\alpha) \):
\[ l = h \sec(\alpha) \]

### Step 4: Calculate the Curved Surface Area
The formula for the curved surface area \( S \) of a cone is given by:
\[ S = \pi r l \]
Substituting the expressions for \( r \) and \( l \):
\[ S = \pi (h \tan(\alpha))(h \sec(\alpha)) \]
\[ S = \pi h^2 \tan(\alpha) \sec(\alpha) \]

### Final Expression
Thus, the area of the curved surface \( S \) of the cone is:
\[ S = \pi h^2 \tan(\alpha) \sec(\alpha) \]

### Summary
The area of the curved surface of a right circular cone with height \( h \) and semi-vertical angle \( \alpha \) is given by:
\[ S = \pi h^2 \tan(\alpha) \sec(\alpha) \]