If the height and base of a triangle are 6 cm and 5 cm respectively, and the area of the triangle is equal to a circle, then find its radius.

Options

A. `sqrt((15)/(pi))cm`
B. `(10)/(pi)cm`
C. `(24)/(pi)cm`
D. `sqrt((12)/(pi))cm`

Answer

To solve the problem step by step, we will calculate the area of the triangle and then use that to find the radius of the circle.

### Step 1: Calculate the area of the triangle
The formula for the area of a triangle is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Given:
- Base = 5 cm
- Height = 6 cm

Substituting the values into the formula:
\[ \text{Area} = \frac{1}{2} \times 5 \, \text{cm} \times 6 \, \text{cm} \]
\[ \text{Area} = \frac{1}{2} \times 30 \, \text{cm}^2 = 15 \, \text{cm}^2 \]

### Step 2: Set the area of the triangle equal to the area of the circle
We know that the area of the triangle is equal to the area of the circle. The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Setting the area of the triangle equal to the area of the circle:
\[ \pi r^2 = 15 \, \text{cm}^2 \]

### Step 3: Solve for the radius \( r \)
To find \( r \), we rearrange the equation:
\[ r^2 = \frac{15}{\pi} \]
Taking the square root of both sides:
\[ r = \sqrt{\frac{15}{\pi}} \]

### Final Answer
Thus, the radius \( r \) of the circle is:
\[ r = \sqrt{\frac{15}{\pi}} \, \text{cm} \]

---