A right angled `triangle` ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. The volume of the solid so obtained is

Options

A. 200 `pi cm^3`
B. 211 `pi cm^3`
C. 100 `pi cm^3`
D. 101 `pi cm^3`

Answer

To find the volume of the solid obtained by revolving the right-angled triangle ABC around the side measuring 12 cm, we can follow these steps:

### Step 1: Identify the dimensions of the triangle
The sides of the right-angled triangle ABC are given as:
- AB = 5 cm (perpendicular side)
- AC = 12 cm (base)
- BC = 13 cm (hypotenuse)

### Step 2: Determine the shape formed by the revolution
When the triangle is revolved around the side AC (12 cm), it forms a cone. In this case:
- The height (h) of the cone is equal to the length of side AC, which is 12 cm.
- The radius (r) of the cone is equal to the length of side AB, which is 5 cm.

### Step 3: Use the formula for the volume of a cone
The formula for the volume (V) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of r and h into the formula:
- r = 5 cm
- h = 12 cm

### Step 4: Calculate the volume
Substituting the values into the volume formula:
\[ V = \frac{1}{3} \pi (5)^2 (12) \]
Calculating \( (5)^2 \):
\[ (5)^2 = 25 \]
Now substitute back into the volume formula:
\[ V = \frac{1}{3} \pi (25) (12) \]
Calculating \( 25 \times 12 \):
\[ 25 \times 12 = 300 \]
Now substitute this value back:
\[ V = \frac{1}{3} \pi (300) \]
Calculating \( \frac{300}{3} \):
\[ \frac{300}{3} = 100 \]
Thus, we have:
\[ V = 100 \pi \, \text{cm}^3 \]

### Final Answer
The volume of the solid obtained by revolving the triangle is:
\[ \text{Volume} = 100 \pi \, \text{cm}^3 \]

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