A sector of central angle `120^(@)` and a radius of 21 cm were made into a cone . Find the height of the cone ( in cm ).

Answer

To find the height of the cone formed from a sector with a central angle of \(120^\circ\) and a radius of \(21\) cm, we will follow these steps:

### Step 1: Calculate the area of the sector
The area of a sector is given by the formula:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
where \(\theta\) is the central angle in degrees and \(r\) is the radius.

Substituting the values:
\[ \text{Area of sector} = \frac{120}{360} \times \pi \times (21)^2 \]
\[ = \frac{1}{3} \times \pi \times 441 \]
\[ = \frac{441\pi}{3} = 147\pi \text{ cm}^2 \]

### Step 2: Relate the area of the sector to the curved surface area of the cone
When the sector is folded to form a cone, the area of the sector becomes the curved surface area of the cone:
\[ \text{Curved Surface Area of Cone} = \pi r l \]
where \(r\) is the base radius of the cone and \(l\) is the slant height.

### Step 3: Find the base radius of the cone
The length of the arc of the sector becomes the circumference of the base of the cone. The length of the arc is given by:
\[ \text{Length of arc} = \frac{\theta}{360} \times 2\pi r \]
Substituting the values:
\[ \text{Length of arc} = \frac{120}{360} \times 2\pi \times 21 \]
\[ = \frac{1}{3} \times 2\pi \times 21 = 14\pi \text{ cm} \]
This length is equal to the circumference of the base of the cone:
\[ 2\pi r = 14\pi \]
Dividing both sides by \(2\pi\):
\[ r = 7 \text{ cm} \]

### Step 4: Use the area of the sector to find the slant height
Now we can use the area of the sector to find the slant height \(l\):
\[ 147\pi = \pi \times 7 \times l \]
Dividing both sides by \(\pi\):
\[ 147 = 7l \]
Dividing both sides by \(7\):
\[ l = 21 \text{ cm} \]

### Step 5: Use the Pythagorean theorem to find the height of the cone
In the cone, we can use the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 21^2 = 7^2 + h^2 \]
\[ 441 = 49 + h^2 \]
Subtracting \(49\) from both sides:
\[ h^2 = 441 - 49 = 392 \]
Taking the square root of both sides:
\[ h = \sqrt{392} = 14\sqrt{2} \text{ cm} \approx 19.8 \text{ cm} \]

### Final Answer
The height of the cone is \(14\sqrt{2} \text{ cm}\) or approximately \(19.8 \text{ cm}\).

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