If the radius of the circum-circle of an isosceles triangle ABC is equal to `AB (= AC)`, then angle A is

Options

A. `pi/6`
B. `pi/3`
C. `pi/2`
D. `(2pi)/3`

Answer

To solve the problem, we need to find the angle A in an isosceles triangle ABC where the radius of the circumcircle (R) is equal to the lengths of the equal sides (AB = AC).

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Let AB = AC = R (the radius of the circumcircle).
- Let angle A be the angle opposite side BC.

2. **Use the Sine Rule**:
- According to the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
- Here, a = BC, b = AC = R, and c = AB = R.

3. **Set Up the Sine Rule for the Isosceles Triangle**:
- Since AB = AC = R, we can write:
\[ \frac{R}{\sin B} = 2R \quad \text{and} \quad \frac{R}{\sin C} = 2R \]
- This simplifies to:
\[ \sin B = \frac{1}{2} \quad \text{and} \quad \sin C = \frac{1}{2} \]

4. **Determine Angles B and C**:
- The sine of an angle is 1/2 at 30 degrees:
\[ B = 30^\circ \quad \text{and} \quad C = 30^\circ \]

5. **Calculate Angle A**:
- The sum of angles in a triangle is 180 degrees:
\[ A + B + C = 180^\circ \]
- Substituting the values of B and C:
\[ A + 30^\circ + 30^\circ = 180^\circ \]
- Simplifying gives:
\[ A + 60^\circ = 180^\circ \]
- Therefore:
\[ A = 180^\circ - 60^\circ = 120^\circ \]

6. **Convert Angle A to Radians**:
- To express angle A in radians:
\[ A = \frac{120 \times \pi}{180} = \frac{2\pi}{3} \]

### Final Result:
Thus, angle A is \( 120^\circ \) or \( \frac{2\pi}{3} \) radians.