In circle NN, N, O, equals, 2NO=2 and the length of arc, O, P, equals, one third, pi OP ⌢

Let's break down the information and solve this step by step.

Given:

• $N, O$ represents the radius of the circle, and it is given that $2NO = 2$, which means the radius $NO = 1$.
• The length of the arc $OP$ is given as $\frac{1}{3} \pi$.

Step 1: Find the central angle

The formula for the length of an arc is given by:

$\text{Arc length} = r \times \theta$

where $r$ is the radius and $\theta$ is the central angle in radians.

We are given that the arc length is $\frac{1}{3} \pi$ and the radius is $1$. So,

$\frac{1}{3} \pi = 1 \times \theta$

Thus, the central angle $\theta$ is $\frac{1}{3} \pi$.

Step 2: Find the area of the sector

The area $A$ of a sector of a circle is given by the formula:

$A = \frac{1}{2} r^2 \theta$

Since the radius $r = 1$ and the angle $\theta = \frac{1}{3} \pi$, we can substitute these values into the formula:

$A = \frac{1}{2} (1)^2 \times \frac{1}{3} \pi = \frac{1}{6} \pi$

Final Answer:

The area of the shaded sector is $\frac{1}{6} \pi$.

---

Would you like further clarification on any part of the solution? Here are some related questions to expand the discussion:

1. How does the formula for the area of a sector change when the radius is different?
2. How would the area of the sector change if the central angle were doubled?
3. What is the relationship between the length of an arc and the area of a sector?
4. How do you calculate the area of a sector if the central angle is given in degrees instead of radians?
5. Can you calculate the area of the entire circle if the radius is 1?

Tip: Always remember that the angle in the formula for arc length and sector area must be in radians, not degrees.