Find the area of the iron sheet required to prepare a cone 20 cm high with base radius 21 cm.

Answer

To find the area of the iron sheet required to prepare a cone with a height of 20 cm and a base radius of 21 cm, we need to calculate the total surface area of the cone. The formula for the total surface area (TSA) of a cone is given by:

\[ \text{TSA} = \pi r (r + l) \]

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone,
- \( \pi \) is a constant approximately equal to \( \frac{22}{7} \).

### Step 1: Find the slant height \( l \)

To find the slant height \( l \), we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the radius and the height of the cone.

\[ l = \sqrt{r^2 + h^2} \]

where:
- \( r = 21 \) cm (radius),
- \( h = 20 \) cm (height).

Calculating \( l \):

\[ l = \sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29 \text{ cm} \]

### Step 2: Substitute the values into the TSA formula

Now that we have \( l = 29 \) cm, we can substitute \( r \) and \( l \) into the TSA formula.

\[ \text{TSA} = \pi r (r + l) = \frac{22}{7} \times 21 \times (21 + 29) \]

Calculating \( (21 + 29) \):

\[ 21 + 29 = 50 \]

Now substituting back into the TSA formula:

\[ \text{TSA} = \frac{22}{7} \times 21 \times 50 \]

### Step 3: Calculate the TSA

Calculating \( 21 \times 50 \):

\[ 21 \times 50 = 1050 \]

Now substituting this value back into the TSA formula:

\[ \text{TSA} = \frac{22}{7} \times 1050 \]

Calculating \( \frac{22 \times 1050}{7} \):

\[ 22 \times 1050 = 23100 \]

Now dividing by 7:

\[ \frac{23100}{7} = 3300 \text{ cm}^2 \]

### Conclusion

Therefore, the area of the iron sheet required to prepare the cone is:

\[ \text{Area} = 3300 \text{ cm}^2 \]

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