If a triangle of base 6 cm has the same area as that of a circle of radius 6 cm, then the altitude of the triangle is

Options

A. `6 pi` cm
B. `8 pi`cm
C. `10 pi`cm
D. `12 pi` cm

Answer

To solve the problem, we need to find the altitude of a triangle that has the same area as a circle with a radius of 6 cm. Let's break this down step by step.

### Step-by-Step Solution:

**Step 1: Calculate the area of the circle.**
The formula for the area of a circle is given by:
\[ \text{Area of Circle} = \pi \times r^2 \]
where \( r \) is the radius. Here, the radius \( r = 6 \) cm.

Substituting the value of the radius:
\[ \text{Area of Circle} = \pi \times (6)^2 = \pi \times 36 = 36\pi \text{ cm}^2 \]

**Step 2: Set up the equation for the area of the triangle.**
The formula for the area of a triangle is:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base of the triangle is given as 6 cm, and we need to find the height (altitude), which we will denote as \( h \).

Substituting the base into the formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times 6 \times h = 3h \text{ cm}^2 \]

**Step 3: Set the areas equal to each other.**
Since the area of the triangle is equal to the area of the circle, we can set the two equations equal:
\[ 3h = 36\pi \]

**Step 4: Solve for the height \( h \).**
To find \( h \), we divide both sides of the equation by 3:
\[ h = \frac{36\pi}{3} = 12\pi \text{ cm} \]

### Final Answer:
The altitude of the triangle is \( 12\pi \) cm.

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