Find the area of the major sector of a circle with a radius of 4cm and angle of 30 

To find the area of the major sector of a circle, we need to:

1. Find the area of the whole circle.
2. Find the area of the minor sector (using the given central angle).
3. Subtract the area of the minor sector from the area of the whole circle to get the area of the major sector.

### Step-by-step solution:

#### 1. Area of the whole circle:
The formula for the area of a circle is:
\[
A_{\text{circle}} = \pi r^2
\]
Where \( r = 4 \, \text{cm} \) (the radius of the circle).
\[
A_{\text{circle}} = \pi \times 4^2 = 16\pi \, \text{cm}^2
\]

#### 2. Area of the minor sector:
The area of a sector is proportional to its central angle. The formula is:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \times A_{\text{circle}}
\]
Here, the central angle \( \theta = 30^\circ \), and the area of the whole circle is \( 16\pi \, \text{cm}^2 \).
\[
A_{\text{minor sector}} = \frac{30^\circ}{360^\circ} \times 16\pi = \frac{1}{12} \times 16\pi = \frac{4\pi}{3} \, \text{cm}^2
\]

#### 3. Area of the major sector:
The major sector is the remaining part of the circle, which has an angle of \( 360^\circ - 30^\circ = 330^\circ \).
The area of the major sector can also be calculated as:
\[
A_{\text{major sector}} = A_{\text{circle}} - A_{\text{minor sector}}
\]
\[
A_{\text{major sector}} = 16\pi - \frac{4\pi}{3} = \frac{48\pi}{3} - \frac{4\pi}{3} = \frac{44\pi}{3} \, \text{cm}^2
\]

Thus, the area of the major sector is:
\[
\boxed{\frac{44\pi}{3} \, \text{cm}^2}
\]
or approximately \( 46.07 \, \text{cm}^2 \).

Would you like more details or further clarification on any part of this process?

### Related Questions:
1. What is the area of the minor sector of a circle with radius 4 cm and a central angle of 30°?
2. How do you find the length of the arc for the major sector in this problem?
3. What is the perimeter of the major sector of the circle?
4. How would the area change if the radius were doubled?
5. How do you calculate the area of a sector when given the arc length?

**Tip:** Always ensure the angle for the sector is in degrees (or radians) before applying the sector area formula.

Find the area of the major sector of a circle with a radius of 4 cm and angle of 30°.

Learn how to calculate the area of the major sector of a circle with a radius of 4 cm and a central angle of 30°. This solution covers key geometric concepts, including circle area and sector area formulas, and is ideal for students in Grades 9-10.

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