There is an equilateral triangle with side 4 and a circle with the centre on one of the vertex of that triangle. The arc of that circle divides the triangle into two parts of equal area. How long is the radius of the circle?

Explanation

To find the radius of the circle that divides an equilateral triangle of side 4 cm into two equal areas, we can follow these steps:

### Step 1: Calculate the area of the equilateral triangle
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:

\[ A = \frac{\sqrt{3}}{4} a^2 \]

Substituting \( a = 4 \):

\[ A = \frac{\sqrt{3}}{4} \times 4^2 = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \text{ cm}^2 \]

### Step 2: Set up the equation for the area divided by the circle
Since the arc of the circle divides the triangle into two equal areas, each part will have an area of:

\[ \text{Area of each part} = \frac{A}{2} = \frac{4\sqrt{3}}{2} = 2\sqrt{3} \text{ cm}^2 \]

### Step 3: Express the area of the sector formed by the circle
The area of the sector formed by the circle at the vertex of the triangle can be expressed as:

\[ \text{Area of sector} = \frac{1}{2} r^2 \theta \]

Where \( \theta \) is the angle in radians. For an equilateral triangle, the angle at each vertex is \( 60^\circ \), which is equivalent to:

\[ \theta = \frac{\pi}{3} \text{ radians} \]

Thus, the area of the sector becomes:

\[ \text{Area of sector} = \frac{1}{2} r^2 \cdot \frac{\pi}{3} = \frac{\pi r^2}{6} \]

### Step 4: Set the area of the sector equal to the area of one part of the triangle
Since the area of the sector is equal to the area of one part of the triangle, we can set up the equation:

\[ \frac{\pi r^2}{6} = 2\sqrt{3} \]

### Step 5: Solve for \( r^2 \)
Multiplying both sides by 6 gives:

\[ \pi r^2 = 12\sqrt{3} \]

Now, divide both sides by \( \pi \):

\[ r^2 = \frac{12\sqrt{3}}{\pi} \]

### Step 6: Calculate \( r \)
Taking the square root of both sides gives:

\[ r = \sqrt{\frac{12\sqrt{3}}{\pi}} = \frac{2\sqrt{3}}{\sqrt{\pi}} \cdot \sqrt{3} = \frac{6}{\sqrt{\pi}} \text{ cm} \]

### Final Answer
The radius of the circle is:

\[ r = \frac{6}{\sqrt{\pi}} \text{ cm} \]