Question

If semi-vertical angle of a cone of radius 7 cm is \(30^{\circ}\), then the curved surface area of the cone (in sq cm) is : (Take \(\pi = \frac{22}{7}\))

• A. 264
• B. 308
• C. 616
• D. 528

Hint:

Remember that for a semi-vertical angle of \(30^{\circ}\), the ratio of radius to slant height is always \(1:2\).
This means the slant height is simply double the radius.
Instantly calculating \(l = 2r = 14\text{ cm}\) saves valuable calculation time during exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from the topic of Mensuration, focusing on the surface area of three-dimensional geometrical figures (specifically a right circular cone).
We are given the radius of the cone's base and its semi-vertical angle.
Our goal is to compute the curved surface area (CSA) of the cone.

Step 2: Key Formula or Approach:
The curved surface area (CSA) of a right circular cone is given by:
\[ \text{CSA} = \pi r l \]
where \(r\) is the radius of the base, and \(l\) is the slant height of the cone.
The relation between the radius \(r\), the slant height \(l\), and the semi-vertical angle \(\alpha\) is:
\[ \sin\alpha = \frac{r}{l} \]

Step 3: Detailed Explanation:
Let the semi-vertical angle of the cone be \(\alpha = 30^{\circ}\).
The radius of the cone is given as \(r = 7\text{ cm}\).
We need to find the slant height \(l\) first.
Using the trigonometric relation:
\[ \sin 30^{\circ} = \frac{r}{l} \]
Substitute the known values:
\[ \frac{1}{2} = \frac{7}{l} \]
Solving for \(l\):
\[ l = 7 \times 2 = 14\text{ cm} \]
Now, calculate the curved surface area (CSA) using the formula:
\[ \text{CSA} = \pi r l \]
Substitute \(\pi = \frac{22}{7}\), \(r = 7\text{ cm}\), and \(l = 14\text{ cm}\):
\[ \text{CSA} = \frac{22}{7} \times 7 \times 14 \]
Cancel the common term \(7\) in the numerator and denominator:
\[ \text{CSA} = 22 \times 14 \]
\[ \text{CSA} = 308\text{ cm}^2 \]

Step 4: Final Answer:
The curved surface area of the cone is \(308\text{ sq cm}\).