https://file.aiquickdraw.com/mathbot/file/1728268617711ydhr7lhy.jpg

Let's go step by step to solve the questions from the image you provided, based on the diagram of a circle with radius \(6 \, \text{cm}\) and the angle \( \angle AOB = 120^\circ \).

### Given:
- Radius of the circle \( r = 6 \, \text{cm} \)
- \( \angle AOB = 120^\circ \)
- \( \pi = 3.14 \)

### a) **Circumference of the circle**

The formula for the circumference \( C \) of a circle is:
\[
C = 2\pi r
\]
Substituting the values:
\[
C = 2 \times 3.14 \times 6 = 37.68 \, \text{cm}
\]

### b) **Area of the circle**

The formula for the area \( A \) of a circle is:
\[
A = \pi r^2
\]
Substituting the values:
\[
A = 3.14 \times 6^2 = 3.14 \times 36 = 113.04 \, \text{cm}^2
\]

### c) **Area of the minor sector OAB**

The formula for the area of a sector is:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta = 120^\circ \) is the central angle.

Substituting the values:
\[
\text{Area of sector} = \frac{120}{360} \times 3.14 \times 36 = \frac{1}{3} \times 113.04 = 37.68 \, \text{cm}^2
\]

### d) **Area of the triangle AOB**

To find the area of triangle \( AOB \), we can use the formula for the area of a triangle with two sides and the included angle:
\[
\text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta)
\]
where \( r = 6 \, \text{cm} \) and \( \theta = 120^\circ \).

Using \( \sin(120^\circ) = \sin(60^\circ) = 0.866 \):
\[
\text{Area of triangle} = \frac{1}{2} \times 6^2 \times 0.866 = \frac{1}{2} \times 36 \times 0.866 = 15.588 \, \text{cm}^2
\]

### e) **Area of the shaded region**

The shaded region is the difference between the area of the minor sector and the area of the triangle:
\[
\text{Area of shaded region} = \text{Area of sector} - \text{Area of triangle}
\]
Substituting the values:
\[
\text{Area of shaded region} = 37.68 - 15.588 = 22.092 \, \text{cm}^2
\]

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Would you like more details or have any further questions?

### Related Questions:
1. How would the circumference change if the radius doubled?
2. What is the length of the arc AB in the minor sector?
3. Can you calculate the area of the major sector of the circle?
4. What would be the area of triangle AOB if the angle were \(90^\circ\)?
5. How does the sine rule apply when finding the area of a triangle in this context?

### Tip:
When solving for the area of a sector, remember to always ensure that the angle is in degrees or radians, and match the formula accordingly!

The diagram below shows a circle of radius 6 cm. AOB = 120°. [Use π = 3.14] Calculate: a) the circumference of the circle, b) the area of the circle, c) the area of the minor sector OAB, d) the area of the triangle AOB, e) the area of the shaded region.

Learn how to calculate the circumference, area of the circle, area of the minor sector, area of the triangle, and the shaded region in a circle of radius 6 cm, with angle AOB = 120°. A step-by-step solution for middle and high school students (Grades 8-10).

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